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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | r19.2zb 4501* | A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4500. 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 4502* | Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.28z 4503* | 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 4504* | Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) |
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.9rzv 4505* | Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.) |
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) | ||
Theorem | r19.28zv 4506* | 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 4507* | 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 4508* | 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 4509* | 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 4510* | 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 4511* | 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 4512* | 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 4513* | Idempotent law for restricted quantifier. Weak version of ralidm 4517, which does not require ax-10 2138, ax-12 2174, but requires ax-8 2107. (Contributed by GG, 30-Sep-2024.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rzal 4514* | Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2813, ax-8 2107. (Revised by GG, 2-Sep-2024.) |
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | ||
Theorem | rzalALT 4515* | Alternate proof of rzal 4514. Shorter, but requiring df-clel 2813, ax-8 2107. (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 4516* | Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2813, ax-8 2107. (Revised by GG, 2-Sep-2024.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅) | ||
Theorem | ralidm 4517 | 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 4518 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2813, ax-8 2107. (Revised by GG, 2-Sep-2024.) |
⊢ ∀𝑥 ∈ ∅ 𝜑 | ||
Theorem | ralf0 4519* | 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 2813, ax-8 2107. (Revised by GG, 2-Sep-2024.) |
⊢ ¬ 𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) | ||
Theorem | ralnralall 4520* | 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 4521* | 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 4522* | 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 4523* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | sbss 4524* | 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 4525 | 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 4526* | Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4522. 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 4527* | Lemma for 2reu4 4528. (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 4528* | Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2652. (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 4529 | 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 4531). It is the analogue for classes of the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1063). | ||
Syntax | cif 4530 | Extend class notation to include the conditional operator for classes. |
class if(𝜑, 𝐴, 𝐵) | ||
Definition | df-if 4531* |
Definition of the conditional operator for classes. The expression
if(𝜑,
𝐴, 𝐵) is read "if 𝜑 then
𝐴
else 𝐵". See
iftrue 4536 and iffalse 4539 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 4588. (Contributed by NM, 15-May-1999.) |
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | ||
Theorem | dfif2 4532* | An alternate definition of the conditional operator df-if 4531 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 4533* | An alternate definition of the conditional operator df-if 4531 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | ||
Theorem | ifeq1 4534 | 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 4535 | 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 4536 | 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 4537 | Inference associated with iftrue 4536. (Contributed by BJ, 7-Oct-2018.) |
⊢ 𝜑 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴 | ||
Theorem | iftrued 4538 | 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 4539 | 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 4540 | Inference associated with iffalse 4539. (Contributed by BJ, 7-Oct-2018.) |
⊢ ¬ 𝜑 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵 | ||
Theorem | iffalsed 4541 | 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 4542 | 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 4539 directly in this case. It happens, e.g., in oevn0 8551. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷) | ||
Theorem | ifsb 4543 | 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 4544* | Alternate definition of the conditional operator df-if 4531. 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 4545* | Alternate definition of the conditional operator df-if 4531. 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 4546* | Alternate definition of the conditional operator df-if 4531. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 4388). (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 4547 | 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 4548 | 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 4549 | 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 4550 | 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 4551 | 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 4552 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) | ||
Theorem | ifbid 4553 | 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 4554 | 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 4555 | 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 4556 | 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 4557 | 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 4558 | 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 4559 | Deduction form of nfif 4560. (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 4560 | 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 4561 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | ||
Theorem | ifeq2da 4562 | Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) | ||
Theorem | ifeq12da 4563 | 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 4564 | 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 4565 | 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 4566 | 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 4567 | 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 4568 | 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 4569 | 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 4570 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
⊢ if(𝜑, 𝐴, 𝐴) = 𝐴 | ||
Theorem | eqif 4571 | 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 4572 | Another expression of the value of the if predicate, analogous to eqif 4571. See also the more specialized iftrue 4536 and iffalse 4539. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶))) | ||
Theorem | elif 4573 | Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.) |
⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶))) | ||
Theorem | ifel 4574 | Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.) |
⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶))) | ||
Theorem | ifcl 4575 | Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶) | ||
Theorem | ifcld 4576 | 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 4577 | Inference associated with ifcl 4575. Membership (closure) of a conditional operator. Also usable to keep a membership hypothesis for the weak deduction theorem dedth 4588 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 4578 | 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 4579 | 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 4580 | Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ if(𝜑, 𝐴, 𝐵) ∈ V | ||
Theorem | ifeqor 4581 | 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 4582 | 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 4583 | 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 4584 | 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 4585 | 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 4586 | 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 4587 | 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 1082. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 1082. 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 4588 | 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 4595. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4601 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html 4601. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒)) & ⊢ 𝜒 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth2h 4589 | Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4592 but requires that each hypothesis have exactly one class variable. See also comments in dedth 4588. (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 4590 | Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4589. (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 4591 | Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4589. (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 4592 | 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 4589 is simpler to use. See also comments in dedth 4588. (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 4593 | Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4592. (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 4594 | Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4592. (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 4595 | Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4588. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓)) & ⊢ 𝜒 ⇒ ⊢ 𝜓 | ||
Theorem | elimhyp2v 4596 | 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 4597 | 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 4598 | Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4588). (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 4599 | 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 4600 | Version of elimhyp 4595 where the hypothesis is deduced from the final antecedent. See divalg 16436 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 |
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