P. 35 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3401-3500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	0ss 3401	The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
⊢ ∅ ⊆ 𝐴
Theorem	ss0b 3402	Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
Theorem	ss0 3403	Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
Theorem	sseq0 3404	A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)
Theorem	ssn0 3405	A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐵 ≠ ∅)
Theorem	abf 3406	A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
⊢ ¬ 𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = ∅
Theorem	eq0rdv 3407*	Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = ∅)
Theorem	csbprc 3408	The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
Theorem	un00 3409	Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)
Theorem	vss 3410	Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
Theorem	disj 3411*	Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
Theorem	disjr 3412*	Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴)
Theorem	disj1 3413*	Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
Theorem	reldisj 3414	Two ways of saying that two classes are disjoint, using the complement of 𝐵 relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ⊆ 𝐶 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
Theorem	disj3 3415	Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
Theorem	disjne 3416	Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ≠ 𝐷)
Theorem	disjel 3417	A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵)
Theorem	disj2 3418	Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
Theorem	ssdisj 3419	Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)
Theorem	undisj1 3420	The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
⊢ (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐶) = ∅) ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ∅)
Theorem	undisj2 3421	The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝐴 ∩ 𝐶) = ∅) ↔ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ∅)
Theorem	ssindif0im 3422	Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)
Theorem	inelcm 3423	The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅)
Theorem	minel 3424	A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ (𝐶 ∩ 𝐵) = ∅) → ¬ 𝐴 ∈ 𝐶)
Theorem	undif4 3425	Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∩ 𝐶) = ∅ → (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶))
Theorem	disjssun 3426	Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ 𝐶))
Theorem	ssdif0im 3427	Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐵) = ∅)
Theorem	vdif0im 3428	Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
⊢ (𝐴 = V → (V ∖ 𝐴) = ∅)
Theorem	difrab0eqim 3429*	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 Jim Kingdon, 3-Aug-2018.)
⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑} → (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅)
Theorem	inssdif0im 3430	Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 → (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)
Theorem	difid 3431	The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
⊢ (𝐴 ∖ 𝐴) = ∅
Theorem	difidALT 3432	The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3431. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∖ 𝐴) = ∅
Theorem	dif0 3433	The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∖ ∅) = 𝐴
Theorem	0dif 3434	The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
⊢ (∅ ∖ 𝐴) = ∅
Theorem	disjdif 3435	A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
Theorem	difin0 3436	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.)
⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅
Theorem	undif1ss 3437	Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐴 ∪ 𝐵)
Theorem	undif2ss 3438	Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ (𝐴 ∪ 𝐵)
Theorem	undifabs 3439	Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴
Theorem	inundifss 3440	The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴
Theorem	disjdif2 3441	The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴)
Theorem	difun2 3442	Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)
Theorem	undifss 3443	Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) ⊆ 𝐵)
Theorem	ssdifin0 3444	A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅)
Theorem	ssdifeq0 3445	A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
Theorem	ssundifim 3446	A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) → (𝐴 ∖ 𝐵) ⊆ 𝐶)
Theorem	difdifdirss 3447	Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) ⊆ ((𝐴 ∖ 𝐶) ∖ (𝐵 ∖ 𝐶))
Theorem	uneqdifeqim 3448	Two ways that 𝐴 and 𝐵 can "partition" 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 → (𝐶 ∖ 𝐴) = 𝐵))
Theorem	r19.2m 3449*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1617). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	r19.2mOLD 3450*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1617). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) Obsolete version of r19.2m 3449 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	r19.3rm 3451*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28m 3452*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.3rmv 3453*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
Theorem	r19.9rmv 3454*	Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
Theorem	r19.28mv 3455*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.45mv 3456*	Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.44mv 3457*	Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)))
Theorem	r19.27m 3458*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
Theorem	r19.27mv 3459*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
Theorem	rzal 3460*	Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexn0 3461*	Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3462). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
Theorem	rexm 3462*	Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
Theorem	ralidm 3463*	Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	ral0 3464	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
⊢ ∀𝑥 ∈ ∅ 𝜑
Theorem	rgenm 3465*	Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	ralf0 3466*	The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
⊢ ¬ 𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
Theorem	ralm 3467	Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	raaanlem 3468*	Special case of raaan 3469 where 𝐴 is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
Theorem	raaan 3469*	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 3470*	Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
Theorem	sbss 3471*	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 3472	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	dcun 3473	The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)
⊢ (𝜑 → DECID 𝑘 ∈ 𝐴)    &   ⊢ (𝜑 → DECID 𝑘 ∈ 𝐵)    ⇒   ⊢ (𝜑 → DECID 𝑘 ∈ (𝐴 ∪ 𝐵))
2.1.15  Conditional operator
Syntax	cif 3474	Extend class notation to include the conditional operator. See df-if 3475 for a description. (In older databases this was denoted "ded".)
class if(𝜑, 𝐴, 𝐵)
Definition	df-if 3475*	Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if 𝜑 then 𝐴 else 𝐵." See iftrue 3479 and iffalse 3482 for its values. In mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise." In the absence of excluded middle, this will tend to be useful where 𝜑 is decidable (in the sense of df-dc 820). (Contributed by NM, 15-May-1999.)
⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}
Theorem	dfif6 3476*	An alternate definition of the conditional operator df-if 3475 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})
Theorem	ifeq1 3477	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 3478	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 3479	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 3480	Inference associated with iftrue 3479. (Contributed by BJ, 7-Oct-2018.)
⊢ 𝜑    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴
Theorem	iftrued 3481	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 3482	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 3483	Inference associated with iffalse 3482. (Contributed by BJ, 7-Oct-2018.)
⊢ ¬ 𝜑    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = 𝐵
Theorem	iffalsed 3484	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 3485	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 3482 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (𝐴 ≠ 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
Theorem	ifsbdc 3486	Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ (DECID 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
Theorem	dfif3 3487*	Alternate definition of the conditional operator df-if 3475. 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	ifeq12 3488	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 3489	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 3490	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 3491	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 3492	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Theorem	ifbid 3493	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 3494	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 3495	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 3496	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 3497	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 3498	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 3499	Deduction version of nfif 3500. (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 3500	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)