P. 46 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4501-4600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	r19.27zv 4501*	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 4502*	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	ralidmw 4503*	Idempotent law for restricted quantifier. Weak version of ralidm 4507, which does not require ax-10 2130, ax-12 2167, but requires ax-8 2101. (Contributed by Gino Giotto, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rzal 4504*	Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2806, ax-8 2101. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rzalALT 4505*	Alternate proof of rzal 4504. Shorter, but requiring df-clel 2806, ax-8 2101. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexn0 4506*	Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2806, ax-8 2101. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
Theorem	ralidm 4507	Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) Reduce axiom usage. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ral0 4508	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2806, ax-8 2101. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ ∀𝑥 ∈ ∅ 𝜑
Theorem	ralf0 4509*	The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) Avoid df-clel 2806, ax-8 2101. (Revised by Gino Giotto, 2-Sep-2024.)
⊢ ¬ 𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
Theorem	rexn0OLD 4510*	Obsolete version of rexn0 4506 as of 2-Sep-2024. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
Theorem	ralidmOLD 4511*	Obsolete version of ralidm 4507 as of 2-Sep-2024. (Contributed by NM, 28-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ral0OLD 4512	Obsolete version of ral0 4508 as of 2-Sep-2024. (Contributed by NM, 20-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥 ∈ ∅ 𝜑
Theorem	ralf0OLD 4513*	Obsolete version of ralf0 4509 as of 2-Sep-2024. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
Theorem	ralnralall 4514*	A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝜑) → 𝜓))
Theorem	falseral0 4515*	A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)
⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)
Theorem	raaan 4516*	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 4517*	Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	sbss 4518*	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 4519	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))
Theorem	raaan2 4520*	Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4516. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓)))
Theorem	2reu4lem 4521*	Lemma for 2reu4 4522. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))
Theorem	2reu4 4522*	Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2646. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	csbdif 4523	Distribution of class substitution over difference of two classes. (Contributed by ML, 14-Jul-2020.)
⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶)
2.1.15  The conditional operator for classes This subsection introduces the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4525). It is the analogue for classes of the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1062).
Syntax	cif 4524	Extend class notation to include the conditional operator for classes.
class if(𝜑, 𝐴, 𝐵)
Definition	df-if 4525*	Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4530 and iffalse 4533 for its values. In the 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". An important use for us is in conjunction with the weak deduction theorem, which is described in the next section, beginning at dedth 4582. (Contributed by NM, 15-May-1999.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
Theorem	dfif2 4526*	An alternate definition of the conditional operator df-if 4525 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 4527*	An alternate definition of the conditional operator df-if 4525 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
Theorem	ifeq1 4528	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 4529	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 4530	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 4531	Inference associated with iftrue 4530. (Contributed by BJ, 7-Oct-2018.)
⊢ 𝜑    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴
Theorem	iftrued 4532	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 4533	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 4534	Inference associated with iffalse 4533. (Contributed by BJ, 7-Oct-2018.)
⊢ ¬ 𝜑    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵
Theorem	iffalsed 4535	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 4536	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 versus applying iffalse 4533 directly in this case. It happens, e.g., in oevn0 8529. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
Theorem	ifsb 4537	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 4538*	Alternate definition of the conditional operator df-if 4525. 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 4539*	Alternate definition of the conditional operator df-if 4525. 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 4540*	Alternate definition of the conditional operator df-if 4525. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 4374). (Contributed by Gérard Lang, 18-Aug-2013.)
⊢ 𝐶 = {𝑥 ∣ 𝜑}    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐵) ∪ (((𝐴 ∖ 𝐵) ∩ 𝐶) ∪ ((𝐵 ∖ 𝐴) ∩ (V ∖ 𝐶))))
Theorem	ifssun 4541	A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
Theorem	ifeq12 4542	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 4543	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 4544	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 4545	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 4546	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Theorem	ifbid 4547	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 4548	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 4549	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 4550	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 4551	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 4552	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 4553	Deduction form of nfif 4554. (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 4554	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 4555	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Theorem	ifeq2da 4556	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Theorem	ifeq12da 4557	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 4558	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 4559	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 4560	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 4561	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 4562	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 4563	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 4564	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
⊢ if(𝜑, 𝐴, 𝐴) = 𝐴
Theorem	eqif 4565	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 4566	Another expression of the value of the if predicate, analogous to eqif 4565. See also the more specialized iftrue 4530 and iffalse 4533. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶)))
Theorem	elif 4567	Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))
Theorem	ifel 4568	Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶)))
Theorem	ifcl 4569	Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶)
Theorem	ifcld 4570	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	ifcli 4571	Inference associated with ifcl 4569. Membership (closure) of a conditional operator. Also usable to keep a membership hypothesis for the weak deduction theorem dedth 4582 when the special case 𝐵 ∈ 𝐶 is provable. (Contributed by NM, 14-Aug-1999.) (Proof shortened by BJ, 1-Sep-2022.)
⊢ 𝐴 ∈ 𝐶    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝐶
Theorem	ifexd 4572	Existence of the conditional operator (deduction form). (Contributed by SN, 26-Jul-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)
Theorem	ifexg 4573	Existence of the conditional operator (closed form). (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
Theorem	ifex 4574	Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) ∈ V
Theorem	ifeqor 4575	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 4576	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 4577	Rewrite a conjunction in a conditional 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 4578	Rewrite a disjunction in a conditional 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 4579	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 4580	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 4581	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)
2.1.16  The weak deduction theorem for set theory This subsection contains a few results related to the weak deduction theorem in set theory. For the weak deduction theorem in propositional calculus, see the section beginning with elimh 1081. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 1081. 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 is 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.)
Theorem	dedth 4582	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 hypothesis eliminated with elimhyp 4589. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4595 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html 4595. (Contributed by NM, 15-May-1999.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒))    &   ⊢ 𝜒    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	dedth2h 4583	Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4586 but requires that each hypothesis have exactly one class variable. See also comments in dedth 4582. (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 4584	Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4583. (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 4585	Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4583. (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 4586	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 4583 is simpler to use. See also comments in dedth 4582. (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 4587	Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4586. (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 4588	Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4586. (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 4589	Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4582. (Contributed by NM, 15-May-1999.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓))    &   ⊢ 𝜒    ⇒   ⊢ 𝜓
Theorem	elimhyp2v 4590	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 4591	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 4592	Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4582). (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 4593	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
Theorem	elimdhyp 4594	Version of elimhyp 4589 where the hypothesis is deduced from the final antecedent. See divalg 16373 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒))    &   ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃 ↔ 𝜒))    &   ⊢ 𝜃    ⇒   ⊢ 𝜒
Theorem	keephyp 4595	Transform a hypothesis 𝜓 that we want to keep (but contains the same class variable 𝐴 used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))    &   ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))    &   ⊢ 𝜓    &   ⊢ 𝜒    ⇒   ⊢ 𝜃
Theorem	keephyp2v 4596	Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4582). (Contributed by NM, 16-Apr-2005.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒))    &   ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃))    &   ⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂))    &   ⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃))    &   ⊢ 𝜓    &   ⊢ 𝜏    ⇒   ⊢ 𝜃
Theorem	keephyp3v 4597	Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜌 ↔ 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))    &   ⊢ 𝜌    &   ⊢ 𝜂    ⇒   ⊢ 𝜏
2.1.17  Power classes
Syntax	cpw 4598	Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class 𝒫 𝐴
Theorem	pwjust 4599*	Soundness justification theorem for df-pw 4600. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴}
Definition	df-pw 4600*	Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if 𝐴 = {3, 5, 7}, then 𝒫 𝐴 = {∅, {3}, {5}, {7}, {3, 5}, {3, 7}, {5, 7}, {3, 5, 7}} (ex-pw 30232). We will later introduce the Axiom of Power Sets ax-pow 5359, which can be expressed in class notation per pwexg 5372. Still later we will prove, in hashpw 14421, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 24-Jun-1993.)
⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}