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

Theorem cdeqab 2981
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 2977 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32abbidv 2397 . 2  |-  ( x  =  y  ->  { z  |  ph }  =  { z  |  ps } )
43cdeqi 2976 1  |- CondEq ( x  =  y  ->  { z  |  ph }  =  { z  |  ps } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   {cab 2269  CondEqwcdeq 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-cdeq 2975
  Copyright terms: Public domain W3C validator