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Theorem List for Metamath Proof Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdisj3 4401 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))

Theoremdisjne 4402 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)

Theoremdisjeq0 4403 Two disjoint sets are equal iff both are empty. (Contributed by AV, 19-Jun-2022.)
((𝐴𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))

Theoremdisjel 4404 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)

Theoremdisj2 4405 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))

Theoremdisj4 4406 Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
((𝐴𝐵) = ∅ ↔ ¬ (𝐴𝐵) ⊊ 𝐴)

Theoremssdisj 4407 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.)
((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)

Theoremdisjpss 4408 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
(((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴𝐵))

Theoremundisj1 4409 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
(((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)

Theoremundisj2 4410 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
(((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)

Theoremssindif0 4411 Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
(𝐴𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Theoreminelcm 4412 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
((𝐴𝐵𝐴𝐶) → (𝐵𝐶) ≠ ∅)

Theoremminel 4413 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.) (Proof shortened by JJ, 14-Jul-2021.)
((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)

Theoremundif4 4414 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))

Theoremdisjssun 4415 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))

Theoremvdif0 4416 Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)
(𝐴 = V ↔ (V ∖ 𝐴) = ∅)

Theoremdifrab0eq 4417* If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
((𝑉 ∖ {𝑥𝑉𝜑}) = ∅ ↔ 𝑉 = {𝑥𝑉𝜑})

Theorempssnel 4418* A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)
(𝐴𝐵 → ∃𝑥(𝑥𝐵 ∧ ¬ 𝑥𝐴))

Theoremdisjdif 4419 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
(𝐴 ∩ (𝐵𝐴)) = ∅

Theoremdifin0 4420 The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∖ 𝐵) = ∅

Theoremunvdif 4421 The union of a class and its complement is the universe. Theorem 5.1(5) of [Stoll] p. 17. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∪ (V ∖ 𝐴)) = V

Theoremundif1 4422 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4419). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)

Theoremundif2 4423 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 4419). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
(𝐴 ∪ (𝐵𝐴)) = (𝐴𝐵)

Theoremundifabs 4424 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴

Theoreminundif 4425 The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴

Theoremdisjdif2 4426 The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)

Theoremdifun2 4427 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Theoremundif 4428 Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) = 𝐵)

Theoremssdifin0 4429 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 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)

Theoremssdifeq0 4430 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
(𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)

Theoremssundif 4431 A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)
(𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)

Theoremdifcom 4432 Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
((𝐴𝐵) ⊆ 𝐶 ↔ (𝐴𝐶) ⊆ 𝐵)

Theorempssdifcom1 4433 Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
((𝐴𝐶𝐵𝐶) → ((𝐶𝐴) ⊊ 𝐵 ↔ (𝐶𝐵) ⊊ 𝐴))

Theorempssdifcom2 4434 Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
((𝐴𝐶𝐵𝐶) → (𝐵 ⊊ (𝐶𝐴) ↔ 𝐴 ⊊ (𝐶𝐵)))

Theoremdifdifdir 4435 Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Theoremuneqdifeq 4436 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.)
((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))

Theoremraldifeq 4437* Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)
(𝜑𝐴𝐵)    &   (𝜑 → ∀𝑥 ∈ (𝐵𝐴)𝜓)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))

Theoremr19.2z 4438* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1974). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)

Theoremr19.2zb 4439* A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4438. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
(𝐴 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))

Theoremr19.3rz 4440* Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
𝑥𝜑       (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Theoremr19.28z 4441* 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.)
𝑥𝜑       (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Theoremr19.3rzv 4442* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
(𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Theoremr19.9rzv 4443* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
(𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥𝐴 𝜑))

Theoremr19.28zv 4444* 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 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Theoremr19.37zv 4445* 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 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃𝑥𝐴 𝜓)))

Theoremr19.45zv 4446* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝐴 𝜓)))

Theoremr19.44zv 4447* Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))

Theoremr19.27z 4448* 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.)
𝑥𝜓       (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))

Theoremr19.27zv 4449* 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 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))

Theoremr19.36zv 4450* 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 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))

Theoremrzal 4451* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = ∅ → ∀𝑥𝐴 𝜑)

Theoremrexn0 4452* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
(∃𝑥𝐴 𝜑𝐴 ≠ ∅)

Theoremralidm 4453* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
(∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)

Theoremral0 4454 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝜑

Theoremralf0 4455* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.)
¬ 𝜑       (∀𝑥𝐴 𝜑𝐴 = ∅)

Theoremralnralall 4456* A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
(𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑) → 𝜓))

Theoremfalseral0 4457* A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)
((∀𝑥 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)

Theoremraaan 4458* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremraaanv 4459* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremsbss 4460* 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]𝑥𝐴𝑦𝐴)

Theoremsbcssg 4461 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremraaan2 4462* Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4458. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
𝑦𝜑    &   𝑥𝜓       ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))

Theorem2reu4lem 4463* Lemma for 2reu4 4464. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))

Theorem2reu4 4464* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2735. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

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 4466). It is the analogue for classes of the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1057).

Syntaxcif 4465 Extend class notation to include the conditional operator for classes.
class if(𝜑, 𝐴, 𝐵)

Definitiondf-if 4466* Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4471 and iffalse 4474 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 4521. (Contributed by NM, 15-May-1999.)

if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}

Theoremdfif2 4467* An alternate definition of the conditional operator df-if 4466 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}

Theoremdfif6 4468* An alternate definition of the conditional operator df-if 4466 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})

Theoremifeq1 4469 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))

Theoremifeq2 4470 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))

Theoremiftrue 4471 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)

Theoremiftruei 4472 Inference associated with iftrue 4471. (Contributed by BJ, 7-Oct-2018.)
𝜑       if(𝜑, 𝐴, 𝐵) = 𝐴

Theoremiftrued 4473 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝜒)       (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴)

Theoremiffalse 4474 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)

Theoremiffalsei 4475 Inference associated with iffalse 4474. (Contributed by BJ, 7-Oct-2018.)
¬ 𝜑       if(𝜑, 𝐴, 𝐵) = 𝐵

Theoremiffalsed 4476 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)

Theoremifnefalse 4477 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 4474 directly in this case. It happens, e.g., in oevn0 8132. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)

Theoremifsb 4478 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
(if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)    &   (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)       𝐶 = if(𝜑, 𝐷, 𝐸)

Theoremdfif3 4479* Alternate definition of the conditional operator df-if 4466. 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 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))

Theoremdfif4 4480* Alternate definition of the conditional operator df-if 4466. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
𝐶 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))

Theoremdfif5 4481* Alternate definition of the conditional operator df-if 4466. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 4333). (Contributed by Gérard Lang, 18-Aug-2013.)
𝐶 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∩ 𝐶) ∪ ((𝐵𝐴) ∩ (V ∖ 𝐶))))

Theoremifeq12 4482 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

Theoremifeq1d 4483 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Theoremifeq2d 4484 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Theoremifeq12d 4485 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))

Theoremifbi 4486 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Theoremifbid 4487 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
(𝜑 → (𝜓𝜒))       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))

Theoremifbieq1d 4488 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))

Theoremifbieq2i 4489 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝜓)    &   𝐴 = 𝐵       if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Theoremifbieq2d 4490 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))

Theoremifbieq12i 4491 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
(𝜑𝜓)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

Theoremifbieq12d 4492 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Theoremnfifd 4493 Deduction form of nfif 4494. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥if(𝜓, 𝐴, 𝐵))

Theoremnfif 4494 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵       𝑥if(𝜑, 𝐴, 𝐵)

Theoremifeq1da 4495 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑𝜓) → 𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Theoremifeq2da 4496 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Theoremifeq12da 4497 Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Theoremifbieq12d2 4498 Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)
(𝜑 → (𝜓𝜒))    &   ((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Theoremifclda 4499 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑𝜓) → 𝐴𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Theoremifeqda 4500 Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)

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