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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | eqreu 2701* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (x = B → (φ ↔ ψ)) ⇒ ⊢ ((B ∈ A ∧ ψ ∧ ∀x ∈ A (φ → x = B)) → ∃!x ∈ A φ) | ||
Theorem | rmo4 2702* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃*x ∈ A φ ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y)) | ||
Theorem | reu4 2703* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y))) | ||
Theorem | reu7 2704* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃x ∈ A ∀y ∈ A (ψ → x = y))) | ||
Theorem | reu8 2705* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃!x ∈ A φ ↔ ∃x ∈ A (φ ∧ ∀y ∈ A (ψ → x = y))) | ||
Theorem | reueq 2706* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
⊢ (B ∈ A ↔ ∃!x ∈ A x = B) | ||
Theorem | rmoan 2707 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
⊢ (∃*x ∈ A φ → ∃*x ∈ A (ψ ∧ φ)) | ||
Theorem | rmoim 2708 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∀x ∈ A (φ → ψ) → (∃*x ∈ A ψ → ∃*x ∈ A φ)) | ||
Theorem | rmoimia 2709 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (x ∈ A → (φ → ψ)) ⇒ ⊢ (∃*x ∈ A ψ → ∃*x ∈ A φ) | ||
Theorem | rmoimi2 2710 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ ∀x((x ∈ A ∧ φ) → (x ∈ B ∧ ψ)) ⇒ ⊢ (∃*x ∈ B ψ → ∃*x ∈ A φ) | ||
Theorem | 2reuswapdc 2711* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
⊢ (DECID ∃x∃y(x ∈ A ∧ (y ∈ B ∧ φ)) → (∀x ∈ A ∃*y ∈ B φ → (∃!x ∈ A ∃y ∈ B φ → ∃!y ∈ B ∃x ∈ A φ))) | ||
Theorem | reuind 2712* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
⊢ (x = y → (φ ↔ ψ)) & ⊢ (x = y → A = B) ⇒ ⊢ ((∀x∀y(((A ∈ 𝐶 ∧ φ) ∧ (B ∈ 𝐶 ∧ ψ)) → A = B) ∧ ∃x(A ∈ 𝐶 ∧ φ)) → ∃!z ∈ 𝐶 ∀x((A ∈ 𝐶 ∧ φ) → z = A)) | ||
Theorem | 2rmorex 2713* | Double restricted quantification with "at most one," analogous to 2moex 1959. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
⊢ (∃*x ∈ A ∃y ∈ B φ → ∀y ∈ B ∃*x ∈ A φ) | ||
Theorem | nelrdva 2714* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |
⊢ ((φ ∧ x ∈ A) → x ≠ B) ⇒ ⊢ (φ → ¬ B ∈ A) | ||
This is a very useless definition, which "abbreviates" (x = y → φ) as CondEq(x = y → φ). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific 3-ary operation (x = y → φ). This is all used as part of a metatheorem: we want to say that ⊢ (x = y → (φ(x) ↔ φ(y))) and ⊢ (x = y → A(x) = A(y)) are provable, for any expressions φ(x) or A(x) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables assumption: every variable in φ(x) is assumed disjoint from x except x itself. For such a proof by induction, we must consider each of the possible forms of φ(x). If it is a variable other than x, then we have CondEq(x = y → A = A) or CondEq(x = y → (φ ↔ φ)), which is provable by cdeqth 2719 and reflexivity. Since we are only working with class and wff expressions, it can't be x itself in set.mm, but if it was we'd have to also prove CondEq(x = y → x = y) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to x or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that y is disjoint from φ(x) and vice versa, because this is maintained through the induction. This is how we satisfy the DV assumptions of cdeqab1 2724 and cdeqab 2722. | ||
Syntax | wcdeq 2715 | Extend wff notation to include conditional equality. This is a technical device used in the proof that Ⅎ is the not-free predicate, and that definitions are conservative as a result. |
wff CondEq(x = y → φ) | ||
Definition | df-cdeq 2716 | Define conditional equality. All the notation to the left of the ↔ is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEqxyφ. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (CondEq(x = y → φ) ↔ (x = y → φ)) | ||
Theorem | cdeqi 2717 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (x = y → φ) ⇒ ⊢ CondEq(x = y → φ) | ||
Theorem | cdeqri 2718 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → φ) ⇒ ⊢ (x = y → φ) | ||
Theorem | cdeqth 2719 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ φ ⇒ ⊢ CondEq(x = y → φ) | ||
Theorem | cdeqnot 2720 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → (φ ↔ ψ)) ⇒ ⊢ CondEq(x = y → (¬ φ ↔ ¬ ψ)) | ||
Theorem | cdeqal 2721* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → (φ ↔ ψ)) ⇒ ⊢ CondEq(x = y → (∀zφ ↔ ∀zψ)) | ||
Theorem | cdeqab 2722* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → (φ ↔ ψ)) ⇒ ⊢ CondEq(x = y → {z ∣ φ} = {z ∣ ψ}) | ||
Theorem | cdeqal1 2723* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → (φ ↔ ψ)) ⇒ ⊢ CondEq(x = y → (∀xφ ↔ ∀yψ)) | ||
Theorem | cdeqab1 2724* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → (φ ↔ ψ)) ⇒ ⊢ CondEq(x = y → {x ∣ φ} = {y ∣ ψ}) | ||
Theorem | cdeqim 2725 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → (φ ↔ ψ)) & ⊢ CondEq(x = y → (χ ↔ θ)) ⇒ ⊢ CondEq(x = y → ((φ → χ) ↔ (ψ → θ))) | ||
Theorem | cdeqcv 2726 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → x = y) | ||
Theorem | cdeqeq 2727 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → A = B) & ⊢ CondEq(x = y → 𝐶 = 𝐷) ⇒ ⊢ CondEq(x = y → (A = 𝐶 ↔ B = 𝐷)) | ||
Theorem | cdeqel 2728 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ CondEq(x = y → A = B) & ⊢ CondEq(x = y → 𝐶 = 𝐷) ⇒ ⊢ CondEq(x = y → (A ∈ 𝐶 ↔ B ∈ 𝐷)) | ||
Theorem | nfcdeq 2729* | If we have a conditional equality proof, where φ is φ(x) and ψ is φ(y), and φ(x) in fact does not have x free in it according to Ⅎ, then φ(x) ↔ φ(y) unconditionally. This proves that Ⅎxφ is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ & ⊢ CondEq(x = y → (φ ↔ ψ)) ⇒ ⊢ (φ ↔ ψ) | ||
Theorem | nfccdeq 2730* | Variation of nfcdeq 2729 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ ℲxA & ⊢ CondEq(x = y → A = B) ⇒ ⊢ A = B | ||
Theorem | ru 2731 |
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 asserting that A is a set only when it is smaller than some other set B. The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 3838. (Contributed by NM, 7-Aug-1994.) |
⊢ {x ∣ x ∉ x} ∉ V | ||
Syntax | wsbc 2732 | 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 2733 |
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 2757 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 2734 below). Unfortunately, Quine's definition requires a recursive syntactical 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 2734, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2733 in the form of sbc8g 2739. 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 2733 and assert that [A / x]φ is always false when A is a proper class. The related definition df-csb 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 2734 |
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 2733 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 2735 instead of df-sbc 2733. (dfsbcq2 2735 is needed because
unlike Quine we do not overload the df-sb 1619 syntax.) As a consequence of
these theorems, we can derive sbc8g 2739, which is a weaker version of
df-sbc 2733 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 2739, so we will allow direct use of df-sbc 2733. Proper substiution 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 2735 | 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 1619 and substitution for class variables df-sbc 2733. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2734. (Contributed by NM, 31-Dec-2016.) |
⊢ (y = A → ([y / x]φ ↔ [A / x]φ)) | ||
Theorem | sbsbc 2736 | Show that df-sb 1619 and df-sbc 2733 are equivalent when the class term A in df-sbc 2733 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1619 for proofs involving df-sbc 2733. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
⊢ ([y / x]φ ↔ [y / x]φ) | ||
Theorem | sbceq1d 2737 | 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 2738 | 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 | sbc8g 2739 | This is the closest we can get to df-sbc 2733 if we start from dfsbcq 2734 (see its comments) and dfsbcq2 2735. (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 | sbcex 2740 | 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 2741 | Equality theorem for class substitution. Class version of sbequ12 1627. (Contributed by NM, 26-Sep-2003.) |
⊢ (x = A → (φ ↔ [A / x]φ)) | ||
Theorem | sbceq2a 2742 | Equality theorem for class substitution. Class version of sbequ12r 1628. (Contributed by NM, 4-Jan-2017.) |
⊢ (A = x → ([A / x]φ ↔ φ)) | ||
Theorem | spsbc 2743 | 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 1631 and rspsbc 2808. (Contributed by NM, 16-Jan-2004.) |
⊢ (A ∈ 𝑉 → (∀xφ → [A / x]φ)) | ||
Theorem | spsbcd 2744 | 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 1631 and rspsbc 2808. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (φ → A ∈ 𝑉) & ⊢ (φ → ∀xψ) ⇒ ⊢ (φ → [A / x]ψ) | ||
Theorem | sbcth 2745 | 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 2746* | Deduction version of sbcth 2745. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
⊢ (φ → ψ) ⇒ ⊢ ((φ ∧ A ∈ 𝑉) → [A / x]ψ) | ||
Theorem | sbcid 2747 | An identity theorem for substitution. See sbid 1630. (Contributed by Mario Carneiro, 18-Feb-2017.) |
⊢ ([x / x]φ ↔ φ) | ||
Theorem | nfsbc1d 2748 | Deduction version of nfsbc1 2749. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
⊢ (φ → ℲxA) ⇒ ⊢ (φ → Ⅎx[A / x]ψ) | ||
Theorem | nfsbc1 2749 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ ℲxA ⇒ ⊢ Ⅎx[A / x]φ | ||
Theorem | nfsbc1v 2750* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
⊢ Ⅎx[A / x]φ | ||
Theorem | nfsbcd 2751 | Deduction version of nfsbc 2752. (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 2752 | 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 2753* | 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 2754* | 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 2755* | 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 2756* | 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 2757* | 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 2758* | An equivalence for class substitution in the spirit of df-clab 2000. 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 2759 | 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 2760* | 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 2761* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2762.) (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 2762* | 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 2763* | 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 2764* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2765 avoids a disjointness condition on x and 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 2765* | 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 2766* | 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 2767* | 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 2768* | 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 2769 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2664 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 2770* | Substitution applied to an atomic wff. Set theory version of eqsb3 2114. (Contributed by Andrew Salmon, 29-Jun-2011.) |
⊢ (A ∈ 𝑉 → ([A / x]x = B ↔ A = B)) | ||
Theorem | sbcng 2771 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
⊢ (A ∈ 𝑉 → ([A / x] ¬ φ ↔ ¬ [A / x]φ)) | ||
Theorem | sbcimg 2772 | 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 2773 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |
⊢ ([A / x](φ ∧ ψ) ↔ ([A / x]φ ∧ [A / x]ψ)) | ||
Theorem | sbcang 2774 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |
⊢ (A ∈ 𝑉 → ([A / x](φ ∧ ψ) ↔ ([A / x]φ ∧ [A / x]ψ))) | ||
Theorem | sbcor 2775 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |
⊢ ([A / x](φ ∨ ψ) ↔ ([A / x]φ ∨ [A / x]ψ)) | ||
Theorem | sbcorg 2776 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |
⊢ (A ∈ 𝑉 → ([A / x](φ ∨ ψ) ↔ ([A / x]φ ∨ [A / x]ψ))) | ||
Theorem | sbcbig 2777 | 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 | sbcal 2778* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |
⊢ ([A / y]∀xφ ↔ ∀x[A / y]φ) | ||
Theorem | sbcalg 2779* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
⊢ (A ∈ 𝑉 → ([A / y]∀xφ ↔ ∀x[A / y]φ)) | ||
Theorem | sbcex2 2780* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
⊢ ([A / y]∃xφ ↔ ∃x[A / y]φ) | ||
Theorem | sbcexg 2781* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
⊢ (A ∈ 𝑉 → ([A / y]∃xφ ↔ ∃x[A / y]φ)) | ||
Theorem | sbceqal 2782* | A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.) |
⊢ (A ∈ 𝑉 → (∀x(x = A → x = B) → A = B)) | ||
Theorem | sbeqalb 2783* | 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 2784 | 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 2785* | 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 2786 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) |
⊢ (φ ↔ ψ) ⇒ ⊢ ([A / x]φ ↔ [A / x]ψ) | ||
Theorem | eqsbc3r 2787* | eqsbc3 2770 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) |
⊢ (A ∈ B → ([A / x]𝐶 = x ↔ 𝐶 = A)) | ||
Theorem | sbc3ang 2788 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (A ∈ 𝑉 → ([A / x](φ ∧ ψ ∧ χ) ↔ ([A / x]φ ∧ [A / x]ψ ∧ [A / x]χ))) | ||
Theorem | sbcel1gv 2789* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (A ∈ 𝑉 → ([A / x]x ∈ B ↔ A ∈ B)) | ||
Theorem | sbcel2gv 2790* | 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 | sbcimdv 2791* | Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ ((φ ∧ A ∈ 𝑉) → ([A / x]ψ → [A / x]χ)) | ||
Theorem | sbctt 2792 | 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 2793 | 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 2794 | 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 2795* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2793. (Contributed by Alan Sare, 10-Nov-2012.) |
⊢ (A ∈ 𝑉 → ([A / x]φ ↔ φ)) | ||
Theorem | sbc2iegf 2796* | 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 2797* | 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 2798* | 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 2799* | 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 2800* | Lemma for sbccom 2801. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) |
⊢ ([A / x][B / y]φ ↔ [B / y][A / x]φ) |
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