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Theorem cdeqab 2952
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 2948 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32abbidv 2295 . 2  |-  ( x  =  y  ->  { z  |  ph }  =  { z  |  ps } )
43cdeqi 2947 1  |- CondEq ( x  =  y  ->  { z  |  ph }  =  { z  |  ps } )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353   {cab 2163  CondEqwcdeq 2945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-cdeq 2946
This theorem is referenced by: (None)
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