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Type | Label | Description |
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Statement | ||
Theorem | ru 3401 |
Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.
In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as 𝐴 ∈ V, asserted that any collection of sets 𝐴 is a set i.e. belongs to the universe V of all sets. In particular, by substituting {𝑥 ∣ 𝑥 ∉ 𝑥} (the "Russell class") for 𝐴, it asserted {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 4730 asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. However, Zermelo was then faced with a "chicken and egg" problem of how to show 𝐵 is a set, leading him to introduce the set-building axioms of Null Set 0ex 4718, Pairing prex 4836, Union uniex 6851, Power Set pwex 4774, and Infinity omex 8423 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 5890 (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287). Russell himself continued in a different direction, avoiding the paradox with his "theory of types." Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 9180 and Cantor's Theorem canth 6508 are provably false! (See ncanth 6509 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 4709 replaces ax-rep 4699) with ax-sep 4709 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic," J. Symb. Logic 9:1-19 (1944). Under our ZF set theory, every set is a member of the Russell class by elirrv 8387 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv 8390). See ruALT 8391 for an alternate proof of ru 3401 derived from that fact. (Contributed by NM, 7-Aug-1994.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Syntax | wsbc 3402 | Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class 𝐴 for setvar variable 𝑥 in wff 𝜑." |
wff [𝐴 / 𝑥]𝜑 | ||
Definition | df-sbc 3403 |
Define the proper substitution of a class for a set.
When 𝐴 is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3429 for our definition, which always evaluates to true for proper classes. Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 3404 below). For example, if 𝐴 is a proper class, Quine's substitution of 𝐴 for 𝑦 in 0 ∈ 𝑦 evaluates to 0 ∈ 𝐴 rather than our falsehood. (This can be seen by substituting 𝐴, 𝑦, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactic breakdown of 𝜑, and it does not seem possible to express it with a single closed formula. If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 3404, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3403 in the form of sbc8g 3410. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of 𝐴 in every use of this definition) we allow direct reference to df-sbc 3403 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3411 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3404. The related definition df-csb 3500 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | dfsbcq 3404 |
Proper substitution of a class for a set in a wff given equal classes.
This is the essence of the sixth axiom of Frege, specifically Proposition
52 of [Frege1879] p. 50.
This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 3403 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3405 instead of df-sbc 3403. (dfsbcq2 3405 is needed because unlike Quine we do not overload the df-sb 1868 syntax.) As a consequence of these theorems, we can derive sbc8g 3410, which is a weaker version of df-sbc 3403 that leaves substitution undefined when 𝐴 is a proper class. However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3410, so we will allow direct use of df-sbc 3403 after theorem sbc2or 3411 below. Proper substitution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.) |
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑)) | ||
Theorem | dfsbcq2 3405 | This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1868 and substitution for class variables df-sbc 3403. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3404. (Contributed by NM, 31-Dec-2016.) |
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbsbc 3406 | Show that df-sb 1868 and df-sbc 3403 are equivalent when the class term 𝐴 in df-sbc 3403 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1868 for proofs involving df-sbc 3403. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbceq1d 3407 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) | ||
Theorem | sbceq1dd 3408 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) | ||
Theorem | sbceqbid 3409* | Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) | ||
Theorem | sbc8g 3410 | This is the closest we can get to df-sbc 3403 if we start from dfsbcq 3404 (see its comments) and dfsbcq2 3405. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | ||
Theorem | sbc2or 3411* | The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for [𝐴 / 𝑥]𝜑 behavior at proper classes, matching the sbc5 3427 (false) and sbc6 3429 (true) conclusions. This is interesting since dfsbcq 3404 and dfsbcq2 3405 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem does not tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable 𝑦 that 𝜑 or 𝐴 may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.) |
⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | ||
Theorem | sbcex 3412 | By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | ||
Theorem | sbceq1a 3413 | Equality theorem for class substitution. Class version of sbequ12 2097. (Contributed by NM, 26-Sep-2003.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbceq2a 3414 | Equality theorem for class substitution. Class version of sbequ12r 2098. (Contributed by NM, 4-Jan-2017.) |
⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | spsbc 3415 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2341 and rspsbc 3484. (Contributed by NM, 16-Jan-2004.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | ||
Theorem | spsbcd 3416 | Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 2341 and rspsbc 3484. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcth 3417 | A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.) |
⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcthdv 3418* | Deduction version of sbcth 3417. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓) | ||
Theorem | sbcid 3419 | An identity theorem for substitution. See sbid 2100. (Contributed by Mario Carneiro, 18-Feb-2017.) |
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | nfsbc1d 3420 | Deduction version of nfsbc1 3421. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) | ||
Theorem | nfsbc1 3421 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | ||
Theorem | nfsbc1v 3422* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑 | ||
Theorem | nfsbcd 3423 | Deduction version of nfsbc 3424. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | ||
Theorem | nfsbc 3424 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 | ||
Theorem | sbcco 3425* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcco2 3426* | A composition law for class substitution. Importantly, 𝑥 may occur free in the class expression substituted for 𝐴. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) ⇒ ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbc5 3427* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | ||
Theorem | sbc6g 3428* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | ||
Theorem | sbc6 3429* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)) | ||
Theorem | sbc7 3430* | An equivalence for class substitution in the spirit of df-clab 2597. Note that 𝑥 and 𝐴 don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)) | ||
Theorem | cbvsbc 3431 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | cbvsbcv 3432* | Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | ||
Theorem | sbciegft 3433* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3434.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | ||
Theorem | sbciegf 3434* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | ||
Theorem | sbcieg 3435* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | ||
Theorem | sbcie2g 3436* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 3437 avoids a disjointness condition on 𝑥, 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒)) | ||
Theorem | sbcie 3437* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbciedf 3438* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | sbcied 3439* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | sbcied2 3440* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | elrabsf 3441 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3329 has implicit substitution). The hypothesis specifies that 𝑥 must not be a free variable in 𝐵. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑)) | ||
Theorem | eqsbc3 3442* | Substitution applied to an atomic wff. Set theory version of eqsb3 2715. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | sbcng 3443 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbcimg 3444 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) | ||
Theorem | sbcan 3445 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcor 3446 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcbig 3447 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | ||
Theorem | sbcn1 3448 | Move negation in and out of class substitution. One direction of sbcng 3443 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcim1 3449 | Distribution of class substitution over implication. One direction of sbcimg 3444 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcbi1 3450 | Distribution of class substitution over biconditional. One direction of sbcbig 3447 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcbi2 3451 | Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) | ||
Theorem | sbcal 3452* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.) |
⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑) | ||
Theorem | sbcex2 3453* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.) |
⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) | ||
Theorem | sbceqal 3454* | Set theory version of sbeqal1 37620. (Contributed by Andrew Salmon, 28-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) | ||
Theorem | sbeqalb 3455* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |
⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → 𝐴 = 𝐵)) | ||
Theorem | sbcbid 3456 | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbcbidv 3457* | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbcbii 3458 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) | ||
Theorem | eqsbc3r 3459* | eqsbc3 3442 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴)) | ||
Theorem | eqsbc3rOLD 3460* | Obsolete proof of eqsbc3r 3459 as of 7-Jul-2021. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 38097 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴)) | ||
Theorem | sbc3an 3461 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓 ∧ [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbcel1v 3462* | Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵) | ||
Theorem | sbcel2gv 3463* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | sbcel21v 3464* | Class substitution into a membership relation. One direction of sbcel2gv 3463 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵) | ||
Theorem | sbcimdv 3465* | Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1729). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbcimdvOLD 3466* | Obsolete proof of sbcimdv 3465 as of 7-Jul-2021. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) | ||
Theorem | sbctt 3467 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | sbcgf 3468 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | sbc19.21g 3469 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓))) | ||
Theorem | sbcg 3470* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3468. (Contributed by Alan Sare, 10-Nov-2012.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | sbc2iegf 3471* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊 & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) | ||
Theorem | sbc2ie 3472* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) | ||
Theorem | sbc2iedv 3473* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒)) | ||
Theorem | sbc3ie 3474* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓) | ||
Theorem | sbccomlem 3475* | Lemma for sbccom 3476. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) |
⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
Theorem | sbccom 3476* | Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
Theorem | sbcralt 3477* | Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbcrext 3478* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbcrextOLD 3479* | Obsolete proof of sbcrext 3478 as of 7-Jul-2021. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbcralg 3480* | Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | ||
Theorem | sbcrex 3481* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcreu 3482* | Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.) |
⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) | ||
Theorem | sbcabel 3483* | Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.) |
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]{𝑦 ∣ 𝜑} ∈ 𝐵 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ∈ 𝐵)) | ||
Theorem | rspsbc 3484* | Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 2341 and spsbc 3415. See also rspsbca 3485 and rspcsbela 3958. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑)) | ||
Theorem | rspsbca 3485* | Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝜑) → [𝐴 / 𝑥]𝜑) | ||
Theorem | rspesbca 3486* | Existence form of rspsbca 3485. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑) | ||
Theorem | spesbc 3487 | Existence form of spsbc 3415. (Contributed by Mario Carneiro, 18-Nov-2016.) |
⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) | ||
Theorem | spesbcd 3488 | form of spsbc 3415. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → [𝐴 / 𝑥]𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | sbcth2 3489* | A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝑥 ∈ 𝐵 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑) | ||
Theorem | ra4v 3490* | Version of ra4 3491 with a dv condition, requiring fewer axioms. This is stdpc5v 1854 for a restricted domain. (Contributed by BJ, 27-Mar-2020.) |
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | ra4 3491 | Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is the axiom stdpc5 2063 of standard predicate calculus for a restricted domain. See ra4v 3490 for a version requiring fewer axioms. (Contributed by NM, 16-Jan-2004.) (Proof shortened by BJ, 27-Mar-2020.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | rmo2 3492* | Alternate definition of restricted "at most one." Note that ∃*𝑥 ∈ 𝐴𝜑 is not equivalent to ∃𝑦 ∈ 𝐴∀𝑥 ∈ 𝐴(𝜑 → 𝑥 = 𝑦) (in analogy to reu6 3362); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3493. (Contributed by NM, 17-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) | ||
Theorem | rmo2i 3493* | Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rmo3 3494* | Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | ||
Theorem | rmob 3495* | Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒))) | ||
Theorem | rmoi 3496* | Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶) | ||
Theorem | rmob2 3497* | Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.) |
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜑 → 𝑥 ∈ 𝐴) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝑥 = 𝐵 ↔ 𝜒)) | ||
Theorem | rmoi2 3498* | Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.) |
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜑 → 𝑥 ∈ 𝐴) & ⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → 𝑥 = 𝐵) | ||
Syntax | csb 3499 | Extend class notation to include the proper substitution of a class for a set into another class. |
class ⦋𝐴 / 𝑥⦌𝐵 | ||
Definition | df-csb 3500* | Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3402, to prevent ambiguity. Theorem sbcel1g 3939 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsb 3956 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.) |
⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} |
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