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Theorem cdeqal1 2807
Description: Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
cdeqnot.1  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cdeqal1  |- CondEq ( x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Distinct variable groups:    ps, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem cdeqal1
StepHypRef Expression
1 cdeqnot.1 . . . 4  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
21cdeqri 2802 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32cbvalv 1836 . 2  |-  ( A. x ph  <->  A. y ps )
43cdeqth 2803 1  |- CondEq ( x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283  CondEqwcdeq 2799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cdeq 2800
This theorem is referenced by: (None)
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