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Theorem cdeqal1 3144
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 3139 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32cbvalv 1984 . 2  |-  ( A. x ph  <->  A. y ps )
43cdeqth 3140 1  |- CondEq ( x  =  y  ->  ( A. x ph  <->  A. y ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549  CondEqwcdeq 3136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-cdeq 3137
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