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Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempssv 3401 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
 |-  ( A  C.  _V  <->  -.  A  =  _V )
 
Theoremdisj 3402* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B )
 
Theoremdisjr 3403* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  B  -.  x  e.  A )
 
Theoremdisj1 3404* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B ) )
 
Theoremreldisj 3405 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_  C  ->  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( C 
 \  B ) ) )
 
Theoremdisj3 3406 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
 
Theoremdisjne 3407 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  C  e.  A  /\  D  e.  B ) 
 ->  C  =/=  D )
 
Theoremdisjel 3408 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  C  e.  A )  ->  -.  C  e.  B )
 
Theoremdisj2 3409 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V  \  B ) )
 
Theoremdisj4 3410 Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
 |-  ( ( A  i^i  B )  =  (/)  <->  -.  ( A  \  B )  C.  A )
 
Theoremssdisj 3411 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
 |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
 
Theoremdisjpss 3412 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  B  =/=  (/) )  ->  A  C.  ( A  u.  B ) )
 
Theoremundisj1 3413 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
 |-  ( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  u.  B )  i^i 
 C )  =  (/) )
 
Theoremundisj2 3414 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C ) )  =  (/) )
 
Theoremssindif0 3415 Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  C_  B  <->  ( A  i^i  ( _V  \  B ) )  =  (/) )
 
Theoreminelcm 3416 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
 |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( B  i^i  C )  =/=  (/) )
 
Theoremminel 3417 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
 |-  ( ( A  e.  B  /\  ( C  i^i  B )  =  (/) )  ->  -.  A  e.  C )
 
Theoremundif4 3418 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  C )  =  (/)  ->  ( A  u.  ( B  \  C ) )  =  ( ( A  u.  B )  \  C ) )
 
Theoremdisjssun 3419 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  B )  =  (/)  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C ) )
 
Theoremssdif0 3420 Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  C_  B  <->  ( A  \  B )  =  (/) )
 
Theoremvdif0 3421 Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  =  _V  <->  ( _V  \  A )  =  (/) )
 
Theorempssdifn0 3422 A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
 |-  ( ( A  C_  B  /\  A  =/=  B )  ->  ( B  \  A )  =/=  (/) )
 
Theorempssdif 3423 A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  ( A  C.  B  ->  ( B  \  A )  =/=  (/) )
 
Theoremssnelpss 3424 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
 |-  ( A  C_  B  ->  ( ( C  e.  B  /\  -.  C  e.  A )  ->  A  C.  B ) )
 
Theorempssnel 3425* A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)
 |-  ( A  C.  B  ->  E. x ( x  e.  B  /\  -.  x  e.  A )
 )
 
Theoremdifin0ss 3426 Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
 |-  ( ( ( A 
 \  B )  i^i 
 C )  =  (/)  ->  ( C  C_  A  ->  C 
 C_  B ) )
 
Theoreminssdif0 3427 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  i^i  B )  C_  C  <->  ( A  i^i  ( B  \  C ) )  =  (/) )
 
Theoremdifid 3428 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 )  =  (/)
 
TheoremdifidALT 3429 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 3428 suggested by David Abernethy, 17-Jun-2012.) (Contributed by NM, 17-Jun-2012.) (Proof modification is discouraged.)
 |-  ( A  \  A )  =  (/)
 
Theoremdif0 3430 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
 
Theorem0dif 3431 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 )  =  (/)
 
Theoremdisjdif 3432 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  i^i  ( B  \  A ) )  =  (/)
 
Theoremdifin0 3433 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  i^i  B )  \  B )  =  (/)
 
Theoremundifv 3434 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  u.  ( _V  \  A ) )  =  _V
 
Theoremundif1 3435 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3432). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
 |-  ( ( A  \  B )  u.  B )  =  ( A  u.  B )
 
Theoremundif2 3436 Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3432). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
 |-  ( A  u.  ( B  \  A ) )  =  ( A  u.  B )
 
Theoremundifabs 3437 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
 |-  ( A  u.  ( A  \  B ) )  =  A
 
Theoreminundif 3438 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  i^i  B )  u.  ( A 
 \  B ) )  =  A
 
Theoremdifun2 3439 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
 |-  ( ( A  u.  B )  \  B )  =  ( A  \  B )
 
Theoremundif 3440 Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  <->  ( A  u.  ( B 
 \  A ) )  =  B )
 
Theoremssdifin0 3441 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  C_  ( B  \  C )  ->  ( A  i^i  C )  =  (/) )
 
Theoremssdifeq0 3442 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
 |-  ( A  C_  ( B  \  A )  <->  A  =  (/) )
 
Theoremssundif 3443 A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)
 |-  ( A  C_  ( B  u.  C )  <->  ( A  \  B )  C_  C )
 
Theoremdifcom 3444 Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
 |-  ( ( A  \  B )  C_  C  <->  ( A  \  C )  C_  B )
 
Theorempssdifcom1 3445 Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
 |-  ( ( A  C_  C  /\  B  C_  C )  ->  ( ( C 
 \  A )  C.  B 
 <->  ( C  \  B )  C.  A ) )
 
Theorempssdifcom2 3446 Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
 |-  ( ( A  C_  C  /\  B  C_  C )  ->  ( B  C.  ( C  \  A )  <->  A  C.  ( C  \  B ) ) )
 
Theoremdifdifdir 3447 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 ) )
 
Theoremuneqdifeq 3448 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_  C  /\  ( A  i^i  B )  =  (/) )  ->  ( ( A  u.  B )  =  C  <->  ( C  \  A )  =  B ) )
 
Theoremr19.2z 3449* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1759). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)
 |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
 
Theoremr19.2zb 3450* A response to the notion that the condition  A  =/=  (/) can be removed in r19.2z 3449. Interestingly enough,  ph does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
 |-  ( A  =/=  (/)  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
 
Theoremr19.3rz 3451* Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
 |- 
 F/ x ph   =>    |-  ( A  =/=  (/)  ->  ( ph 
 <-> 
 A. x  e.  A  ph ) )
 
Theoremr19.28z 3452* 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.)
 |- 
 F/ x ph   =>    |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 ph  /\  A. x  e.  A  ps ) ) )
 
Theoremr19.3rzv 3453* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
 |-  ( A  =/=  (/)  ->  ( ph 
 <-> 
 A. x  e.  A  ph ) )
 
Theoremr19.9rzv 3454* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
 |-  ( A  =/=  (/)  ->  ( ph 
 <-> 
 E. x  e.  A  ph ) )
 
Theoremr19.28zv 3455* 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  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 ph  /\  A. x  e.  A  ps ) ) )
 
Theoremr19.37zv 3456* 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  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  (
 ph  ->  E. x  e.  A  ps ) ) )
 
Theoremr19.45zv 3457* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  (
 ph  \/  E. x  e.  A  ps ) ) )
 
Theoremr19.27z 3458* 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.)
 |- 
 F/ x ps   =>    |-  ( A  =/=  (/) 
 ->  ( A. x  e.  A  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  ph  /\  ps )
 ) )
 
Theoremr19.27zv 3459* 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  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  ps ) ) )
 
Theoremr19.36zv 3460* 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  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  (
 A. x  e.  A  ph 
 ->  ps ) ) )
 
Theoremrzal 3461* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  (/)  ->  A. x  e.  A  ph )
 
Theoremrexn0 3462* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
 
Theoremralidm 3463* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
 |-  ( A. x  e.  A  A. x  e.  A  ph  <->  A. x  e.  A  ph )
 
Theoremral0 3464 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
 |- 
 A. x  e.  (/)  ph
 
Theoremrgenz 3465* Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.)
 |-  ( ( A  =/=  (/)  /\  x  e.  A )  ->  ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremralf0 3466* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
 |- 
 -.  ph   =>    |-  ( A. x  e.  A  ph  <->  A  =  (/) )
 
Theoremraaan 3467* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x  e.  A  A. y  e.  A  (
 ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  A. y  e.  A  ps ) )
 
Theoremraaanv 3468* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
 |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) )
 
Theoremsbss 3469* 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  C_  A  <->  y 
 C_  A )
 
2.1.15  "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.)

 
Syntaxcif 3470 Extend class notation to include the conditional operator. See df-if 3471 for a description. (In older databases this was denoted "ded".)
 class  if ( ph ,  A ,  B )
 
Definitiondf-if 3471* Define the conditional operator. Read  if ( ph ,  A ,  B ) as "if  ph then  A else  B." See iftrue 3476 and iffalse 3477 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 3511 for the main part of the weak deduction theorem, elimhyp 3518 to eliminate a hypothesis, and keephyp 3524 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 ( ph ,  A ,  B )  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
 
Theoremdfif2 3472* An alternate definition of the conditional operator df-if 3471 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
 |- 
 if ( ph ,  A ,  B )  =  { x  |  ( ( x  e.  B  -> 
 ph )  ->  ( x  e.  A  /\  ph ) ) }
 
Theoremdfif6 3473* An alternate definition of the conditional operator df-if 3471 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
 |- 
 if ( ph ,  A ,  B )  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )
 
Theoremifeq1 3474 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C )
 )
 
Theoremifeq2 3475 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B )
 )
 
Theoremiftrue 3476 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.)
 |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
 
Theoremiffalse 3477 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
 |-  ( -.  ph  ->  if ( ph ,  A ,  B )  =  B )
 
Theoremifnefalse 3478 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 3477 directly in this case. It happens, e.g., in oevn0 6400. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  =/=  B  ->  if ( A  =  B ,  C ,  D )  =  D )
 
Theoremifsb 3479 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
 |-  ( if ( ph ,  A ,  B )  =  A  ->  C  =  D )   &    |-  ( if ( ph ,  A ,  B )  =  B  ->  C  =  E )   =>    |-  C  =  if ( ph ,  D ,  E )
 
Theoremdfif3 3480* Alternate definition of the conditional operator df-if 3471. Note that  ph 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  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  i^i  C )  u.  ( B  i^i  ( _V  \  C ) ) )
 
Theoremdfif4 3481* Alternate definition of the conditional operator df-if 3471. Note that  ph is independent of  x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
 |-  C  =  { x  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  u.  B )  i^i  ( ( A  u.  ( _V  \  C ) )  i^i  ( B  u.  C ) ) )
 
Theoremdfif5 3482* Alternate definition of the conditional operator df-if 3471. Note that  ph is independent of  x i.e. a constant true or false (see also abvor0 3379). (Contributed by Gérard Lang, 18-Aug-2013.)
 |-  C  =  { x  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  i^i  B )  u.  ( ( ( A  \  B )  i^i  C )  u.  ( ( B  \  A )  i^i  ( _V  \  C ) ) ) )
 
Theoremifeq12 3483 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  D ) )
 
Theoremifeq1d 3484 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2d 3485 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
Theoremifeq12d 3486 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  D ) )
 
Theoremifbi 3487 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
 |-  ( ( ph  <->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
 
Theoremifbid 3488 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B ) )
 
Theoremifbieq2i 3489 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  A  =  B   =>    |-  if ( ph ,  C ,  A )  =  if ( ps ,  C ,  B )
 
Theoremifbieq2d 3490 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ch ,  C ,  B ) )
 
Theoremifbieq12i 3491 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
 |-  ( ph  <->  ps )   &    |-  A  =  C   &    |-  B  =  D   =>    |- 
 if ( ph ,  A ,  B )  =  if ( ps ,  C ,  D )
 
Theoremifbieq12d 3492 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D ) )
 
Theoremnfifd 3493 Deduction version of nfif 3494. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x if ( ps ,  A ,  B ) )
 
Theoremnfif 3494 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x if ( ph ,  A ,  B )
 
Theoremifeq1da 3495 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2da 3496 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  -.  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
Theoremifclda 3497 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  ps )  ->  A  e.  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  e.  C )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
 
Theoremcsbifg 3498 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ if ( ph ,  B ,  C )  =  if ( [. A  /  x ]. ph ,  [_ A  /  x ]_ B ,  [_ A  /  x ]_ C ) )
 
Theoremelimif 3499 Elimination of a conditional operator contained in a wff  ps. (Contributed by NM, 15-Feb-2005.)
 |-  ( if ( ph ,  A ,  B )  =  A  ->  ( ps 
 <->  ch ) )   &    |-  ( if ( ph ,  A ,  B )  =  B  ->  ( ps  <->  th ) )   =>    |-  ( ps  <->  ( ( ph  /\ 
 ch )  \/  ( -.  ph  /\  th )
 ) )
 
Theoremifbothda 3500 A wff  th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   &    |-  (
 ( et  /\  ph )  ->  ps )   &    |-  ( ( et 
 /\  -.  ph )  ->  ch )   =>    |-  ( et  ->  th )
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