Home | Metamath
Proof Explorer Theorem List (p. 41 of 424) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-27159) |
Hilbert Space Explorer
(27160-28684) |
Users' Mathboxes
(28685-42360) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | undif 4001 | Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.) |
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | ||
Theorem | ssdifin0 4002 | A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.) |
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅) | ||
Theorem | ssdifeq0 4003 | A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.) |
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) | ||
Theorem | ssundif 4004 | A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.) |
⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶) | ||
Theorem | difcom 4005 | Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.) |
⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | ||
Theorem | pssdifcom1 4006 | Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐶 ∖ 𝐴) ⊊ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊊ 𝐴)) | ||
Theorem | pssdifcom2 4007 | Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵))) | ||
Theorem | difdifdir 4008 | Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.) |
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶)) | ||
Theorem | uneqdifeq 4009 | Two ways to say that 𝐴 and 𝐵 partition 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). (Contributed by FL, 17-Nov-2008.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵)) | ||
Theorem | uneqdifeqOLD 4010 | Obsolete proof of uneqdifeq 4009 as of 14-Jul-2021. (Contributed by FL, 17-Nov-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵)) | ||
Theorem | raldifeq 4011* | Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝜓) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) | ||
Theorem | r19.2z 4012* | Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1879). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.) |
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | r19.2zb 4013* | A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4012. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.) |
⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.3rz 4014* | Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.28z 4015* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | r19.3rzv 4016* | Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) |
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.9rzv 4017* | Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.) |
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.28zv 4018* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.) |
⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | r19.37zv 4019* | Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.) |
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | r19.45zv 4020* | Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))) | ||
Theorem | r19.44zv 4021* | Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))) | ||
Theorem | r19.27z 4022* | Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) | ||
Theorem | r19.27zv 4023* | Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.) |
⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) | ||
Theorem | r19.36zv 4024* | Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.) |
⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))) | ||
Theorem | rzal 4025* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rexn0 4026* | Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) | ||
Theorem | ralidm 4027* | Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | ral0 4028 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
⊢ ∀𝑥 ∈ ∅ 𝜑 | ||
Theorem | rgenzOLD 4029* | Obsolete as of 22-Jul-2021. (Contributed by NM, 8-Dec-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝐴 ≠ ∅ ∧ 𝑥 ∈ 𝐴) → 𝜑) ⇒ ⊢ ∀𝑥 ∈ 𝐴 𝜑 | ||
Theorem | ralf0 4030* | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ¬ 𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) | ||
Theorem | ralf0OLD 4031* | Obsolete proof of ralf0 4030 as of 14-Jul-2021. (Contributed by NM, 26-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ¬ 𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) | ||
Theorem | raaan 4032* | Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | raaanv 4033* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | sbss 4034* | Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
⊢ ([𝑦 / 𝑥]𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴) | ||
Theorem | sbcssg 4035 | Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ⊆ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ⊆ ⦋𝐴 / 𝑥⦌𝐶)) | ||
In a Hilbert system of logic (which consists of a set of axioms, modus ponens, and the generalization rule), converting a deduction to a proof using the Deduction Theorem (taught in introductory logic books) involves an exponential increase of the number of steps as hypotheses are successively eliminated. Here is a trick that is not as general as the Deduction Theorem but requires only a linear increase in the number of steps. The general problem: We want to convert a deduction P |- Q into a proof of the theorem |- P -> Q i.e., we want to eliminate the hypothesis P. Normally this is done using the Deduction (meta)Theorem, which looks at the microscopic steps of the deduction and usually doubles or triples the number of these microscopic steps for each hypothesis that is eliminated. We will look at a special case of this problem, without appealing to the Deduction Theorem. We assume ZF with class notation. A and B are arbitrary (possibly proper) classes. P, Q, R, S and T are wffs. We define the conditional operator, if(P, A, B), as follows: if(P, A, B) =def= { x | (x \in A & P) v (x \in B & -. P) } (where x does not occur in A, B, or P). Lemma 1. A = if(P, A, B) -> (P <-> R), B = if(P, A, B) -> (S <-> R), S |- R Proof: Logic and Axiom of Extensionality. Lemma 2. A = if(P, A, B) -> (Q <-> T), T |- P -> Q Proof: Logic and Axiom of Extensionality. Here's a simple example that illustrates how it works. Suppose we have a deduction Ord A |- Tr A which means, "Assume A is an ordinal class. Then A is a transitive class." Note that A is a class variable that may be substituted with any class expression, so this is really a deduction scheme. We want to convert this to a proof of the theorem (scheme) |- Ord A -> Tr A. The catch is that we must be able to prove "Ord A" for at least one object A (and this is what makes it weaker than the ordinary Deduction Theorem). However, it is easy to prove |- Ord 0 (the empty set is ordinal). (For a typical textbook "theorem," i.e. deduction, there is usually at least one object satisfying each hypothesis, otherwise the theorem would not be very useful. We can always go back to the standard Deduction Theorem for those hypotheses where this is not the case.) Continuing with the example: Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Ord A <-> Ord if(Ord A, A, 0)) (1) |- 0 = if(Ord A, A, 0) -> (Ord 0 <-> Ord if(Ord A, A, 0)) (2) From (1), (2) and |- Ord 0, Lemma 1 yields |- Ord if(Ord A, A, 0) (3) From (3) and substituting if(Ord A, A, 0) for A in the original deduction, |- Tr if(Ord A, A, 0) (4) Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Tr A <-> Tr if(Ord A, A, 0)) (5) From (4) and (5), Lemma 2 yields |- Ord A -> Tr A (Q.E.D.) | ||
Syntax | cif 4036 | Extend class notation to include the conditional operator. See df-if 4037 for a description. (In older databases this was denoted "ded".) |
class if(𝜑, 𝐴, 𝐵) | ||
Definition | df-if 4037* |
Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if
𝜑 then 𝐴 else 𝐵."
See iftrue 4042 and iffalse 4045 for its
values. In mathematical literature, this operator is rarely defined
formally but is implicit in informal definitions such as "let
f(x)=0 if
x=0 and 1/x otherwise." (In older versions of this database, this
operator was denoted "ded" and called the "deduction
class.")
An important use for us is in conjunction with the weak deduction theorem, which converts a hypothesis into an antecedent. In that role, 𝐴 is a class variable in the hypothesis and 𝐵 is a class (usually a constant) that makes the hypothesis true when it is substituted for 𝐴. See dedth 4089 for the main part of the weak deduction theorem, elimhyp 4096 to eliminate a hypothesis, and keephyp 4102 to keep a hypothesis. See the Deduction Theorem link on the Metamath Proof Explorer Home Page for a description of the weak deduction theorem. (Contributed by NM, 15-May-1999.) |
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | ||
Theorem | dfif2 4038* | An alternate definition of the conditional operator df-if 4037 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.) |
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))} | ||
Theorem | dfif6 4039* | An alternate definition of the conditional operator df-if 4037 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | ||
Theorem | ifeq1 4040 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶)) | ||
Theorem | ifeq2 4041 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵)) | ||
Theorem | iftrue 4042 | Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | ||
Theorem | iftruei 4043 | Inference associated with iftrue 4042. (Contributed by BJ, 7-Oct-2018.) |
⊢ 𝜑 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴 | ||
Theorem | iftrued 4044 | Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴) | ||
Theorem | iffalse 4045 | Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | ||
Theorem | iffalsei 4046 | Inference associated with iffalse 4045. (Contributed by BJ, 7-Oct-2018.) |
⊢ ¬ 𝜑 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵 | ||
Theorem | iffalsed 4047 | Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → ¬ 𝜒) ⇒ ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐵) | ||
Theorem | ifnefalse 4048 | When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 4045 directly in this case. It happens, e.g., in oevn0 7482. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷) | ||
Theorem | ifsb 4049 | Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.) |
⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷) & ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ 𝐶 = if(𝜑, 𝐷, 𝐸) | ||
Theorem | dfif3 4050* | Alternate definition of the conditional operator df-if 4037. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 𝐶 = {𝑥 ∣ 𝜑} ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) | ||
Theorem | dfif4 4051* | Alternate definition of the conditional operator df-if 4037. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) |
⊢ 𝐶 = {𝑥 ∣ 𝜑} ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) | ||
Theorem | dfif5 4052* | Alternate definition of the conditional operator df-if 4037. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 3907). (Contributed by Gérard Lang, 18-Aug-2013.) |
⊢ 𝐶 = {𝑥 ∣ 𝜑} ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐵) ∪ (((𝐴 ∖ 𝐵) ∩ 𝐶) ∪ ((𝐵 ∖ 𝐴) ∩ (V ∖ 𝐶)))) | ||
Theorem | ifeq12 4053 | Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷)) | ||
Theorem | ifeq1d 4054 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | ||
Theorem | ifeq2d 4055 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) | ||
Theorem | ifeq12d 4056 | Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷)) | ||
Theorem | ifbi 4057 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) | ||
Theorem | ifbid 4058 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) | ||
Theorem | ifbieq1d 4059 | Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶)) | ||
Theorem | ifbieq2i 4060 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ 𝐴 = 𝐵 ⇒ ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵) | ||
Theorem | ifbieq2d 4061 | Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) | ||
Theorem | ifbieq12i 4062 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷) | ||
Theorem | ifbieq12d 4063 | Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) | ||
Theorem | nfifd 4064 | Deduction version of nfif 4065. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) | ||
Theorem | nfif 4065 | Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵) | ||
Theorem | ifeq1da 4066 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | ||
Theorem | ifeq2da 4067 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) | ||
Theorem | ifeq12da 4068 | Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)) | ||
Theorem | ifbieq12d2 4069 | Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) | ||
Theorem | ifclda 4070 | Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) | ||
Theorem | ifeqda 4071 | Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶) | ||
Theorem | elimif 4072 | Elimination of a conditional operator contained in a wff 𝜓. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.) |
⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃))) | ||
Theorem | ifbothda 4073 | A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ ((𝜂 ∧ 𝜑) → 𝜓) & ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒) ⇒ ⊢ (𝜂 → 𝜃) | ||
Theorem | ifboth 4074 | A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | ifid 4075 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
⊢ if(𝜑, 𝐴, 𝐴) = 𝐴 | ||
Theorem | eqif 4076 | Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.) |
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))) | ||
Theorem | ifval 4077 | Another expression of the value of the if predicate, analogous to eqif 4076. See also the more specialized iftrue 4042 and iffalse 4045. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶))) | ||
Theorem | elif 4078 | Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.) |
⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶))) | ||
Theorem | ifel 4079 | Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.) |
⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶))) | ||
Theorem | ifcl 4080 | Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶) | ||
Theorem | ifcld 4081 | Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) | ||
Theorem | ifeqor 4082 | The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵) | ||
Theorem | ifnot 4083 | Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.) |
⊢ if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴) | ||
Theorem | ifan 4084 | Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) | ||
Theorem | ifor 4085 | Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵)) | ||
Theorem | 2if2 4086 | Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐷 = 𝐴) & ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ 𝜃) → 𝐷 = 𝐵) & ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶) ⇒ ⊢ (𝜑 → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶))) | ||
Theorem | ifcomnan 4087 | Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.) |
⊢ (¬ (𝜑 ∧ 𝜓) → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶))) | ||
Theorem | csbif 4088 | Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | dedth 4089 | Weak deduction theorem that eliminates a hypothesis 𝜑, making it become an antecedent. We assume that a proof exists for 𝜑 when the class variable 𝐴 is replaced with a specific class 𝐵. The hypothesis 𝜒 should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 4096. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4102 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒)) & ⊢ 𝜒 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth2h 4090 | Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4093 but requires that each hypothesis has exactly one class variable. See also comments in dedth 4089. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜒 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜓, 𝐵, 𝐷) → (𝜃 ↔ 𝜏)) & ⊢ 𝜏 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | dedth3h 4091 | Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4090. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃 ↔ 𝜏)) & ⊢ (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏 ↔ 𝜂)) & ⊢ (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂 ↔ 𝜁)) & ⊢ 𝜁 ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | dedth4h 4092 | Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4090. (Contributed by NM, 16-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏 ↔ 𝜂)) & ⊢ (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂 ↔ 𝜁)) & ⊢ (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁 ↔ 𝜎)) & ⊢ (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎 ↔ 𝜌)) & ⊢ 𝜌 ⇒ ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏) | ||
Theorem | dedth2v 4093 | Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 4090 is simpler to use. See also comments in dedth 4089. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) & ⊢ 𝜃 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth3v 4094 | Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4093. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) & ⊢ 𝜏 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth4v 4095 | Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4093. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃 ↔ 𝜏)) & ⊢ (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏 ↔ 𝜂)) & ⊢ 𝜂 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | elimhyp 4096 | Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4089. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓)) & ⊢ 𝜒 ⇒ ⊢ 𝜓 | ||
Theorem | elimhyp2v 4097 | Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜑 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂)) & ⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃)) & ⊢ 𝜏 ⇒ ⊢ 𝜃 | ||
Theorem | elimhyp3v 4098 | Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) & ⊢ (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂 ↔ 𝜁)) & ⊢ (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁 ↔ 𝜎)) & ⊢ (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎 ↔ 𝜏)) & ⊢ 𝜂 ⇒ ⊢ 𝜏 | ||
Theorem | elimhyp4v 4099 | Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4089). (Contributed by NM, 16-Apr-2005.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) & ⊢ (𝐹 = if(𝜑, 𝐹, 𝐺) → (𝜏 ↔ 𝜓)) & ⊢ (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂 ↔ 𝜁)) & ⊢ (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁 ↔ 𝜎)) & ⊢ (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎 ↔ 𝜌)) & ⊢ (𝐺 = if(𝜑, 𝐹, 𝐺) → (𝜌 ↔ 𝜓)) & ⊢ 𝜂 ⇒ ⊢ 𝜓 | ||
Theorem | elimel 4100 | Eliminate a membership hypothesis for weak deduction theorem, when special case 𝐵 ∈ 𝐶 is provable. (Contributed by NM, 15-May-1999.) |
⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ if(𝐴 ∈ 𝐶, 𝐴, 𝐵) ∈ 𝐶 |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |