MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdeqri Unicode version

Theorem cdeqri 2977
Description: Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
cdeqri.1  |- CondEq ( x  =  y  ->  ph )
Assertion
Ref Expression
cdeqri  |-  ( x  =  y  ->  ph )

Proof of Theorem cdeqri
StepHypRef Expression
1 cdeqri.1 . 2  |- CondEq ( x  =  y  ->  ph )
2 df-cdeq 2975 . 2  |-  (CondEq (
x  =  y  ->  ph )  <->  ( x  =  y  ->  ph ) )
31, 2mpbi 199 1  |-  ( x  =  y  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623  CondEqwcdeq 2974
This theorem is referenced by:  cdeqnot  2979  cdeqal  2980  cdeqab  2981  cdeqal1  2982  cdeqab1  2983  cdeqim  2984  cdeqeq  2986  cdeqel  2987  nfcdeq  2988  cdeqbox  25029  cdeqnxt  25030  cdequnt  25031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-cdeq 2975
  Copyright terms: Public domain W3C validator