ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifexd Unicode version

Theorem ifexd 4469
Description: Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
Hypotheses
Ref Expression
ifexd.1  |-  ( ph  ->  A  e.  V )
ifexd.2  |-  ( ph  ->  B  e.  W )
Assertion
Ref Expression
ifexd  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  _V )

Proof of Theorem ifexd
StepHypRef Expression
1 ifexd.1 . . 3  |-  ( ph  ->  A  e.  V )
2 ifexd.2 . . 3  |-  ( ph  ->  B  e.  W )
31, 2ifelpwund 4467 . 2  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  ~P ( A  u.  B ) )
43elexd 2743 1  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141   _Vcvv 2730    u. cun 3119   ifcif 3526   ~Pcpw 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator