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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrabeq0 3301 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
 |-  ( { x  e.  A  |  ph }  =  (/)  <->  A. x  e.  A  -.  ph )
 
Theoremabeq0 3302 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
 |-  ( { x  |  ph
 }  =  (/)  <->  A. x  -.  ph )
 
Theoremrabxmdc 3303* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
 |-  ( A. xDECID  ph  ->  A  =  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } ) )
 
Theoremrabnc 3304* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
 
Theoremun0 3305 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  u.  (/) )  =  A
 
Theoremin0 3306 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  (/) )  =  (/)
 
Theoreminv1 3307 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
 |-  ( A  i^i  _V )  =  A
 
Theoremunv 3308 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
 |-  ( A  u.  _V )  =  _V
 
Theorem0ss 3309 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
 |-  (/)  C_  A
 
Theoremss0b 3310 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  C_  (/)  <->  A  =  (/) )
 
Theoremss0 3311 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
 |-  ( A  C_  (/)  ->  A  =  (/) )
 
Theoremsseq0 3312 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  C_  B  /\  B  =  (/) )  ->  A  =  (/) )
 
Theoremssn0 3313 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  B  =/=  (/) )
 
Theoremabf 3314 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
 |- 
 -.  ph   =>    |- 
 { x  |  ph }  =  (/)
 
Theoremeq0rdv 3315* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
 |-  ( ph  ->  -.  x  e.  A )   =>    |-  ( ph  ->  A  =  (/) )
 
Theoremcsbprc 3316 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  e.  _V 
 ->  [_ A  /  x ]_ B  =  (/) )
 
Theoremun00 3317 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
 |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
 
Theoremvss 3318 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  C_  A  <->  A  =  _V )
 
Theoremdisj 3319* 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 3320* 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 3321* 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 3322 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 3323 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
 
Theoremdisjne 3324 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 3325 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 3326 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V  \  B ) )
 
Theoremssdisj 3327 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 )  =  (/) )
 
Theoremundisj1 3328 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 3329 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 ) )  =  (/) )
 
Theoremssindif0im 3330 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  C_  B  ->  ( A  i^i  ( _V  \  B ) )  =  (/) )
 
Theoreminelcm 3331 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 3332 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 3333 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 3334 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 ) )
 
Theoremssdif0im 3335 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  C_  B  ->  ( A  \  B )  =  (/) )
 
Theoremvdif0im 3336 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
 |-  ( A  =  _V  ->  ( _V  \  A )  =  (/) )
 
Theoremdifrab0eqim 3337* 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  e.  V  |  ph
 }  ->  ( V  \  { x  e.  V  |  ph } )  =  (/) )
 
Theoreminssdif0im 3338 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
 |-  ( ( A  i^i  B )  C_  C  ->  ( A  i^i  ( B 
 \  C ) )  =  (/) )
 
Theoremdifid 3339 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 3340 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 3339. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  \  A )  =  (/)
 
Theoremdif0 3341 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 3342 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 3343 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 3344 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 )  =  (/)
 
Theoremundif1ss 3345 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 )  u.  B )  C_  ( A  u.  B )
 
Theoremundif2ss 3346 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  u.  ( B  \  A ) ) 
 C_  ( A  u.  B )
 
Theoremundifabs 3347 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
 |-  ( A  u.  ( A  \  B ) )  =  A
 
Theoreminundifss 3348 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  i^i  B )  u.  ( A 
 \  B ) ) 
 C_  A
 
Theoremdifun2 3349 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
 |-  ( ( A  u.  B )  \  B )  =  ( A  \  B )
 
Theoremundifss 3350 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( A  C_  B  <->  ( A  u.  ( B 
 \  A ) ) 
 C_  B )
 
Theoremssdifin0 3351 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 3352 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  =  (/) )
 
Theoremssundifim 3353 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  C_  ( B  u.  C )  ->  ( A  \  B ) 
 C_  C )
 
Theoremdifdifdirss 3354 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 ) 
 C_  ( ( A 
 \  C )  \  ( B  \  C ) )
 
Theoremuneqdifeqim 3355 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_  C  /\  ( A  i^i  B )  =  (/) )  ->  ( ( A  u.  B )  =  C  ->  ( C  \  A )  =  B )
 )
 
Theoremr19.2m 3356* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1572). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
 
Theoremr19.3rm 3357* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
 |- 
 F/ x ph   =>    |-  ( E. y  y  e.  A  ->  ( ph 
 <-> 
 A. x  e.  A  ph ) )
 
Theoremr19.28m 3358* 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.)
 |- 
 F/ x ph   =>    |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 ph  /\  A. x  e.  A  ps ) ) )
 
Theoremr19.3rmv 3359* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
 |-  ( E. y  y  e.  A  ->  ( ph 
 <-> 
 A. x  e.  A  ph ) )
 
Theoremr19.9rmv 3360* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |-  ( E. y  y  e.  A  ->  ( ph 
 <-> 
 E. x  e.  A  ph ) )
 
Theoremr19.28mv 3361* 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.)
 |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 ph  /\  A. x  e.  A  ps ) ) )
 
Theoremr19.45mv 3362* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 |-  ( E. x  x  e.  A  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  (
 ph  \/  E. x  e.  A  ps ) ) )
 
Theoremr19.44mv 3363* Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 |-  ( E. y  y  e.  A  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  ( E. x  e.  A  ph 
 \/  ps ) ) )
 
Theoremr19.27m 3364* 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.)
 |- 
 F/ x ps   =>    |-  ( E. x  x  e.  A  ->  (
 A. x  e.  A  ( ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  ps ) ) )
 
Theoremr19.27mv 3365* 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.)
 |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  ps ) ) )
 
Theoremrzal 3366* 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 3367* Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3368). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
 
Theoremrexm 3368* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
 |-  ( E. x  e.  A  ph  ->  E. x  x  e.  A )
 
Theoremralidm 3369* 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 3370 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
 |- 
 A. x  e.  (/)  ph
 
Theoremrgenm 3371* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |-  ( ( E. x  x  e.  A  /\  x  e.  A )  -> 
 ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremralf0 3372* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
 |- 
 -.  ph   =>    |-  ( A. x  e.  A  ph  <->  A  =  (/) )
 
Theoremralm 3373 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
 |-  ( ( E. x  x  e.  A  ->  A. x  e.  A  ph ) 
 <-> 
 A. x  e.  A  ph )
 
Theoremraaanlem 3374* Special case of raaan 3375 where  A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) ) )
 
Theoremraaan 3375* 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 3376* 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 3377* 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 )
 
Theoremsbcssg 3378 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  e.  V  ->  ( [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
 
2.1.15  Conditional operator
 
Syntaxcif 3379 Extend class notation to include the conditional operator. See df-if 3380 for a description. (In older databases this was denoted "ded".)
 class  if ( ph ,  A ,  B )
 
Definitiondf-if 3380* Define the conditional operator. Read  if ( ph ,  A ,  B ) as "if  ph then  A else  B." See iftrue 3384 and iffalse 3387 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  ph is decidable (in the sense of df-dc 779). (Contributed by NM, 15-May-1999.)

 |- 
 if ( ph ,  A ,  B )  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
 
Theoremdfif6 3381* An alternate definition of the conditional operator df-if 3380 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 3382 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 3383 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 3384 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 )
 
Theoremiftruei 3385 Inference associated with iftrue 3384. (Contributed by BJ, 7-Oct-2018.)
 |-  ph   =>    |- 
 if ( ph ,  A ,  B )  =  A
 
Theoremiftrued 3386 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ch )   =>    |-  ( ph  ->  if ( ch ,  A ,  B )  =  A )
 
Theoremiffalse 3387 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
 |-  ( -.  ph  ->  if ( ph ,  A ,  B )  =  B )
 
Theoremiffalsei 3388 Inference associated with iffalse 3387. (Contributed by BJ, 7-Oct-2018.)
 |- 
 -.  ph   =>    |- 
 if ( ph ,  A ,  B )  =  B
 
Theoremiffalsed 3389 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  if ( ch ,  A ,  B )  =  B )
 
Theoremifnefalse 3390 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 3387 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  =/=  B  ->  if ( A  =  B ,  C ,  D )  =  D )
 
Theoremifsbdc 3391 Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
 |-  ( if ( ph ,  A ,  B )  =  A  ->  C  =  D )   &    |-  ( if ( ph ,  A ,  B )  =  B  ->  C  =  E )   =>    |-  (DECID 
 ph  ->  C  =  if ( ph ,  D ,  E ) )
 
Theoremdfif3 3392* Alternate definition of the conditional operator df-if 3380. 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 ) ) )
 
Theoremifeq12 3393 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  D ) )
 
Theoremifeq1d 3394 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2d 3395 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
Theoremifeq12d 3396 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 3397 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
 |-  ( ( ph  <->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
 
Theoremifbid 3398 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B ) )
 
Theoremifbieq1d 3399 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  B ,  C ) )
 
Theoremifbieq2i 3400 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 )
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