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Theorem equsal 1688
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 1639 . 2  |-  ( A. x ( x  =  y  ->  ps )  <->  ( E. x  x  =  y  ->  ps )
)
3 equsal.2 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43pm5.74i 179 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ps )
)
54albii 1429 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  ps ) )
6 a9e 1657 . . 3  |-  E. x  x  =  y
76a1bi 242 . 2  |-  ( ps  <->  ( E. x  x  =  y  ->  ps )
)
82, 5, 73bitr4i 211 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1312   F/wnf 1419   E.wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  intirr  4893  fun11  5158
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