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Theorem cdeqal1 2982
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 . . . . 5  |- CondEq ( x  =  y  ->  ( ph 
<->  ps ) )
21cdeqri 2977 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32cbvalv 1942 . . 3  |-  ( A. x ph  <->  A. y ps )
43a1i 10 . 2  |-  ( x  =  y  ->  ( A. x ph  <->  A. y ps ) )
54cdeqi 2976 1  |- CondEq ( x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527    = wceq 1623  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
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-cdeq 2975
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