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

Theorem cdeqab 2872
Description: Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
cdeqnot.1  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cdeqab  |- CondEq ( x  =  y  ->  { z  |  ph }  =  { z  |  ps } )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem cdeqab
StepHypRef Expression
1 cdeqnot.1 . . . 4  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
21cdeqri 2868 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32abbidv 2235 . 2  |-  ( x  =  y  ->  { z  |  ph }  =  { z  |  ps } )
43cdeqi 2867 1  |- CondEq ( x  =  y  ->  { z  |  ph }  =  { z  |  ps } )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1316   {cab 2103  CondEqwcdeq 2865
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-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-cdeq 2866
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator