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Theorem equsal 1656
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.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
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.1 . . 3  |-  F/ x ps
2119.23 1609 . 2  |-  ( A. x ( x  =  y  ->  ps )  <->  ( E. x  x  =  y  ->  ps )
)
3 equsal.2 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43pm5.74i 178 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ps )
)
54albii 1400 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  ps ) )
6 a9e 1627 . . 3  |-  E. x  x  =  y
76a1bi 241 . 2  |-  ( ps  <->  ( E. x  x  =  y  ->  ps )
)
82, 5, 73bitr4i 210 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   F/wnf 1390   E.wex 1422
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-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  intirr  4741  fun11  4997
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