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Theorem equsal 1913
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
equsal.1  |-  F/ x ps
equsal.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsal  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )

Proof of Theorem equsal
StepHypRef Expression
1 equsal.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2 equsal.1 . . . . . 6  |-  F/ x ps
3219.3 1793 . . . . 5  |-  ( A. x ps  <->  ps )
41, 3syl6bbr 254 . . . 4  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ps )
)
54pm5.74i 236 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  A. x ps ) )
65albii 1556 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  A. x ps )
)
72nfri 1754 . . . . 5  |-  ( ps 
->  A. x ps )
87a1d 22 . . . 4  |-  ( ps 
->  ( x  =  y  ->  A. x ps )
)
92, 8alrimi 1757 . . 3  |-  ( ps 
->  A. x ( x  =  y  ->  A. x ps ) )
10 ax9o 1903 . . 3  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
119, 10impbii 180 . 2  |-  ( ps  <->  A. x ( x  =  y  ->  A. x ps ) )
126, 11bitr4i 243 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   F/wnf 1534
This theorem is referenced by:  equsalh  1914  equsex  1915  dvelimf  1950  sb6x  1982  asymref2  5076  intirr  5077  fun11  5331  pm13.192  27713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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