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Theorem equsal 2138
Description: An equivalence related to implicit substitution. (Contributed by NM, 2-Jun-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 2003 . 2  |-  ( A. x ( x  =  y  ->  ps )  <->  ( E. x  x  =  y  ->  ps )
)
3 equsal.2 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43pm5.74i 253 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ps )
)
54albii 1701 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  ps ) )
6 ax6e 2104 . . 3  |-  E. x  x  =  y
76a1bi 343 . 2  |-  ( ps  <->  ( E. x  x  =  y  ->  ps )
)
82, 5, 73bitr4i 285 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1452   E.wex 1673   F/wnf 1677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678
This theorem is referenced by:  equsalh  2139  equsex  2140  axc9lem2OLD  2144  dvelimf  2178  sb6x  2223  sb6rf  2262  asymref2  5235  intirr  5236  fun11  5669  pm13.192  36804
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