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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | r19.28zv 4501* | 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 4502* | 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 4503* | 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 4504* | 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 4505* | 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 4506* | 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 4507* | 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 4508* | Idempotent law for restricted quantifier. Weak version of ralidm 4512, which does not require ax-10 2141, ax-12 2177, but requires ax-8 2110. (Contributed by GG, 30-Sep-2024.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
| Theorem | rzal 4509* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2816, ax-8 2110. (Revised by GG, 2-Sep-2024.) |
| ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | ||
| Theorem | rzalALT 4510* | Alternate proof of rzal 4509. Shorter, but requiring df-clel 2816, ax-8 2110. (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 4511* | Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2816, ax-8 2110. (Revised by GG, 2-Sep-2024.) |
| ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) | ||
| Theorem | ralidm 4512 | Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) Reduce axiom usage. (Revised by GG, 2-Sep-2024.) |
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
| Theorem | ral0 4513 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2816, ax-8 2110. (Revised by GG, 2-Sep-2024.) |
| ⊢ ∀𝑥 ∈ ∅ 𝜑 | ||
| Theorem | ralf0 4514* | 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 2816, ax-8 2110. (Revised by GG, 2-Sep-2024.) |
| ⊢ ¬ 𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) | ||
| Theorem | ralnralall 4515* | 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 4516* | 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 4517* | 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 4518* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) | ||
| Theorem | sbss 4519* | 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 4520 | 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 4521* | Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4517. 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 4522* | Lemma for 2reu4 4523. (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 4523* | Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2655. (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 4524 | 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) | ||
This subsection introduces the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4526). It is the analogue for classes of the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1064). | ||
| Syntax | cif 4525 | Extend class notation to include the conditional operator for classes. |
| class if(𝜑, 𝐴, 𝐵) | ||
| Definition | df-if 4526* |
Definition of the conditional operator for classes. The expression
if(𝜑,
𝐴, 𝐵) is read "if 𝜑 then
𝐴
else 𝐵". See
iftrue 4531 and iffalse 4534 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 4584. (Contributed by NM, 15-May-1999.) |
| ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | ||
| Theorem | dfif2 4527* | An alternate definition of the conditional operator df-if 4526 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 4528* | An alternate definition of the conditional operator df-if 4526 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
| ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | ||
| Theorem | ifeq1 4529 | 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 4530 | 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 4531 | 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 4532 | Inference associated with iftrue 4531. (Contributed by BJ, 7-Oct-2018.) |
| ⊢ 𝜑 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴 | ||
| Theorem | iftrued 4533 | 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 4534 | 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 4535 | Inference associated with iffalse 4534. (Contributed by BJ, 7-Oct-2018.) |
| ⊢ ¬ 𝜑 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵 | ||
| Theorem | iffalsed 4536 | 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 4537 | 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 4534 directly in this case. It happens, e.g., in oevn0 8553. (Contributed by David A. Wheeler, 15-May-2015.) |
| ⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷) | ||
| Theorem | iftrueb 4538 | When the branches are not equal, an "if" condition results in the first branch if and only if its condition is true. (Contributed by SN, 16-Oct-2025.) |
| ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐴 ↔ 𝜑)) | ||
| Theorem | ifsb 4539 | 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 4540* | Alternate definition of the conditional operator df-if 4526. 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 4541* | Alternate definition of the conditional operator df-if 4526. 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 4542* | Alternate definition of the conditional operator df-if 4526. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 4383). (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 4543 | 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 4544 | 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 4545 | 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 4546 | 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 4547 | 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 4548 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
| ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) | ||
| Theorem | ifbid 4549 | 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 4550 | 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 4551 | 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 4552 | 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 4553 | 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 4554 | 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 4555 | Deduction form of nfif 4556. (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 4556 | 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 4557 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | ||
| Theorem | ifeq2da 4558 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) | ||
| Theorem | ifeq12da 4559 | 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 4560 | 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 4561 | 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 4562 | 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 4563 | 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 4564 | 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 4565 | 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 4566 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
| ⊢ if(𝜑, 𝐴, 𝐴) = 𝐴 | ||
| Theorem | eqif 4567 | 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 4568 | Another expression of the value of the if predicate, analogous to eqif 4567. See also the more specialized iftrue 4531 and iffalse 4534. (Contributed by BJ, 6-Apr-2019.) |
| ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶))) | ||
| Theorem | elif 4569 | Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.) |
| ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶))) | ||
| Theorem | ifel 4570 | Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.) |
| ⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶))) | ||
| Theorem | ifcl 4571 | Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.) |
| ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶) | ||
| Theorem | ifcld 4572 | 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 4573 | Inference associated with ifcl 4571. Membership (closure) of a conditional operator. Also usable to keep a membership hypothesis for the weak deduction theorem dedth 4584 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 4574 | 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 4575 | 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 4576 | Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ if(𝜑, 𝐴, 𝐵) ∈ V | ||
| Theorem | ifeqor 4577 | 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 4578 | 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 4579 | 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 4580 | 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 4581 | 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 4582 | 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 4583 | 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) | ||
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 1083. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 1083. 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 4584 | 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 4591. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4597 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html 4597. (Contributed by NM, 15-May-1999.) |
| ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒)) & ⊢ 𝜒 ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | dedth2h 4585 | Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4588 but requires that each hypothesis have exactly one class variable. See also comments in dedth 4584. (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 4586 | Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4585. (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 4587 | Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4585. (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 4588 | 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 4585 is simpler to use. See also comments in dedth 4584. (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 4589 | Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4588. (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 4590 | Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4588. (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 4591 | Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4584. (Contributed by NM, 15-May-1999.) |
| ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓)) & ⊢ 𝜒 ⇒ ⊢ 𝜓 | ||
| Theorem | elimhyp2v 4592 | 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 4593 | 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 4594 | Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4584). (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 4595 | 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 4596 | Version of elimhyp 4591 where the hypothesis is deduced from the final antecedent. See divalg 16440 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 4597 | 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 4598 | Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4584). (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 4599 | 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 | ||
| Syntax | cpw 4600 | 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 | ||
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