P. 37 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 3601-3700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	disjpss 3601	A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
⊢ (((A ∩ B) = ∅ ∧ B ≠ ∅) → A ⊊ (A ∪ B))
Theorem	undisj1 3602	The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
⊢ (((A ∩ C) = ∅ ∧ (B ∩ C) = ∅) ↔ ((A ∪ B) ∩ C) = ∅)
Theorem	undisj2 3603	The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ (((A ∩ B) = ∅ ∧ (A ∩ C) = ∅) ↔ (A ∩ (B ∪ C)) = ∅)
Theorem	ssindif0 3604	Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
⊢ (A ⊆ B ↔ (A ∩ (V ∖ B)) = ∅)
Theorem	inelcm 3605	The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
⊢ ((A ∈ B ∧ A ∈ C) → (B ∩ C) ≠ ∅)
Theorem	minel 3606	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 3607	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 3608	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	ssdif0 3609	Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
⊢ (A ⊆ B ↔ (A ∖ B) = ∅)
Theorem	vdif0 3610	Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)
⊢ (A = V ↔ (V ∖ A) = ∅)
Theorem	pssdifn0 3611	A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
⊢ ((A ⊆ B ∧ A ≠ B) → (B ∖ A) ≠ ∅)
Theorem	pssdif 3612	A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ (A ⊊ B → (B ∖ A) ≠ ∅)
Theorem	ssnelpss 3613	A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
⊢ (A ⊆ B → ((C ∈ B ∧ ¬ C ∈ A) → A ⊊ B))
Theorem	ssnelpssd 3614	Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3613. (Contributed by David Moews, 1-May-2017.)
⊢ (φ → A ⊆ B)    &   ⊢ (φ → C ∈ B)    &   ⊢ (φ → ¬ C ∈ A)    ⇒   ⊢ (φ → A ⊊ B)
Theorem	pssnel 3615*	A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)
⊢ (A ⊊ B → ∃x(x ∈ B ∧ ¬ x ∈ A))
Theorem	difin0ss 3616	Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
⊢ (((A ∖ B) ∩ C) = ∅ → (C ⊆ A → C ⊆ B))
Theorem	inssdif0 3617	Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
⊢ ((A ∩ B) ⊆ C ↔ (A ∩ (B ∖ C)) = ∅)
Theorem	difid 3618	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 3619	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 3618. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (A ∖ A) = ∅
Theorem	dif0 3620	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 3621	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 3622	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 3623	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	undifv 3624	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
Theorem	undif1 3625	Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3622). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
⊢ ((A ∖ B) ∪ B) = (A ∪ B)
Theorem	undif2 3626	Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3622). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
⊢ (A ∪ (B ∖ A)) = (A ∪ B)
Theorem	undifabs 3627	Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
⊢ (A ∪ (A ∖ B)) = A
Theorem	inundif 3628	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.)
⊢ ((A ∩ B) ∪ (A ∖ B)) = A
Theorem	difun2 3629	Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
⊢ ((A ∪ B) ∖ B) = (A ∖ B)
Theorem	undif 3630	Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)
⊢ (A ⊆ B ↔ (A ∪ (B ∖ A)) = B)
Theorem	ssdifin0 3631	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 3632	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	ssundif 3633	A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)
⊢ (A ⊆ (B ∪ C) ↔ (A ∖ B) ⊆ C)
Theorem	difcom 3634	Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
⊢ ((A ∖ B) ⊆ C ↔ (A ∖ C) ⊆ B)
Theorem	pssdifcom1 3635	Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
⊢ ((A ⊆ C ∧ B ⊆ C) → ((C ∖ A) ⊊ B ↔ (C ∖ B) ⊊ A))
Theorem	pssdifcom2 3636	Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
⊢ ((A ⊆ C ∧ B ⊆ C) → (B ⊊ (C ∖ A) ↔ A ⊊ (C ∖ B)))
Theorem	difdifdir 3637	Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
⊢ ((A ∖ B) ∖ C) = ((A ∖ C) ∖ (B ∖ C))
Theorem	uneqdifeq 3638	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.)
⊢ ((A ⊆ C ∧ (A ∩ B) = ∅) → ((A ∪ B) = C ↔ (C ∖ A) = B))
Theorem	r19.2z 3639*	Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1659). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)
⊢ ((A ≠ ∅ ∧ ∀x ∈ A φ) → ∃x ∈ A φ)
Theorem	r19.2zb 3640*	A response to the notion that the condition A ≠ ∅ can be removed in r19.2z 3639. 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 3641*	Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
⊢ Ⅎxφ    ⇒   ⊢ (A ≠ ∅ → (φ ↔ ∀x ∈ A φ))
Theorem	r19.28z 3642*	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 3643*	Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
⊢ (A ≠ ∅ → (φ ↔ ∀x ∈ A φ))
Theorem	r19.9rzv 3644*	Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
⊢ (A ≠ ∅ → (φ ↔ ∃x ∈ A φ))
Theorem	r19.28zv 3645*	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 3646*	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 3647*	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	r19.27z 3648*	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 3649*	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 3650*	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	rzal 3651*	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 3652*	Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
⊢ (∃x ∈ A φ → A ≠ ∅)
Theorem	ralidm 3653*	Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ)
Theorem	ral0 3654	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
⊢ ∀x ∈ ∅ φ
Theorem	rgenz 3655*	Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.)
⊢ ((A ≠ ∅ ∧ x ∈ A) → φ)    ⇒   ⊢ ∀x ∈ A φ
Theorem	ralf0 3656*	The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
⊢ ¬ φ    ⇒   ⊢ (∀x ∈ A φ ↔ A = ∅)
Theorem	raaan 3657*	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 3658*	Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ))
Theorem	sbss 3659*	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	sbcss 3660	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 ∈ B → ([̣A / x]̣C ⊆ D ↔ [A / x]C ⊆ [A / x]D))
Theorem	sscon34 3661	Contraposition law for subset. (Contributed by SF, 11-Mar-2015.)
⊢ (A ⊆ B ↔ ∼ B ⊆ ∼ A)
2.1.14  "Weak deduction theorem" for set theory 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's 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.)
Syntax	cif 3662	Extend class notation to include the conditional operator. See df-if 3663 for a description. (In older databases this was denoted "ded".)
class if(φ, A, B)
Definition	df-if 3663*	Define the conditional operator. Read if(φ, A, B) as "if φ then A else B." See iftrue 3668 and iffalse 3669 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 older versions of this database, this operator was denoted "ded" and called the "deduction class.") An important use for us is in conjunction with the weak deduction theorem, which converts a hypothesis into an antecedent. In that role, A is a class variable in the hypothesis and B is a class (usually a constant) that makes the hypothesis true when it is substituted for A. See dedth 3703 for the main part of the weak deduction theorem, elimhyp 3710 to eliminate a hypothesis, and keephyp 3716 to keep a hypothesis. See the Deduction Theorem link on the Metamath Proof Explorer Home Page for a description of the weak deduction theorem. (Contributed by NM, 15-May-1999.)
⊢ if(φ, A, B) = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}
Theorem	dfif2 3664*	An alternate definition of the conditional operator df-if 3663 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 3665*	An alternate definition of the conditional operator df-if 3663 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
⊢ if(φ, A, B) = ({x ∈ A ∣ φ} ∪ {x ∈ B ∣ ¬ φ})
Theorem	ifeq1 3666	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 3667	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 3668	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	iffalse 3669	Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
⊢ (¬ φ → if(φ, A, B) = B)
Theorem	ifnefalse 3670	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 vs. applying iffalse 3669 directly in this case. It happens, e.g., in oevn0 in set.mm. (Contributed by David A. Wheeler, 15-May-2015.)
⊢ (A ≠ B → if(A = B, C, D) = D)
Theorem	ifsb 3671	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 3672*	Alternate definition of the conditional operator df-if 3663. 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 3673*	Alternate definition of the conditional operator df-if 3663. 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 3674*	Alternate definition of the conditional operator df-if 3663. Note that φ is independent of x i.e. a constant true or false (see also abvor0 3567). (Contributed by Gérard Lang, 18-Aug-2013.)
⊢ C = {x ∣ φ}    ⇒   ⊢ if(φ, A, B) = ((A ∩ B) ∪ (((A ∖ B) ∩ C) ∪ ((B ∖ A) ∩ (V ∖ C))))
Theorem	ifeq12 3675	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 3676	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 3677	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 3678	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 3679	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
⊢ ((φ ↔ ψ) → if(φ, A, B) = if(ψ, A, B))
Theorem	ifbid 3680	Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → if(ψ, A, B) = if(χ, A, B))
Theorem	ifbieq2i 3681	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 3682	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 3683	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 3684	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 3685	Deduction version of nfif 3686. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ (φ → Ⅎxψ)    &   ⊢ (φ → ℲxA)    &   ⊢ (φ → ℲxB)    ⇒   ⊢ (φ → Ⅎx if(ψ, A, B))
Theorem	nfif 3686	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    ⇒   ⊢ Ⅎx if(φ, A, B)
Theorem	ifeq1da 3687	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((φ ∧ ψ) → A = B)    ⇒   ⊢ (φ → if(ψ, A, C) = if(ψ, B, C))
Theorem	ifeq2da 3688	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((φ ∧ ¬ ψ) → A = B)    ⇒   ⊢ (φ → if(ψ, C, A) = if(ψ, C, B))
Theorem	ifclda 3689	Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((φ ∧ ψ) → A ∈ C)    &   ⊢ ((φ ∧ ¬ ψ) → B ∈ C)    ⇒   ⊢ (φ → if(ψ, A, B) ∈ C)
Theorem	csbifg 3690	Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.)
⊢ (A ∈ V → [A / x] if(φ, B, C) = if([̣A / x]̣φ, [A / x]B, [A / x]C))
Theorem	elimif 3691	Elimination of a conditional operator contained in a wff ψ. (Contributed by NM, 15-Feb-2005.)
⊢ ( if(φ, A, B) = A → (ψ ↔ χ))    &   ⊢ ( if(φ, A, B) = B → (ψ ↔ θ))    ⇒   ⊢ (ψ ↔ ((φ ∧ χ) ∨ (¬ φ ∧ θ)))
Theorem	ifbothda 3692	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 3693	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 3694	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
⊢ if(φ, A, A) = A
Theorem	eqif 3695	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	elif 3696	Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
⊢ (A ∈ if(φ, B, C) ↔ ((φ ∧ A ∈ B) ∨ (¬ φ ∧ A ∈ C)))
Theorem	ifel 3697	Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
⊢ ( if(φ, A, B) ∈ C ↔ ((φ ∧ A ∈ C) ∨ (¬ φ ∧ B ∈ C)))
Theorem	ifcl 3698	Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
⊢ ((A ∈ C ∧ B ∈ C) → if(φ, A, B) ∈ C)
Theorem	ifeqor 3699	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 3700	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)