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Theorem equsal 1999
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 1819 . 2  |-  ( A. x ( x  =  y  ->  ps )  <->  ( E. x  x  =  y  ->  ps )
)
3 equsal.2 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43pm5.74i 237 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  ps )
)
54albii 1575 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  ps ) )
6 a9e 1952 . . 3  |-  E. x  x  =  y
76a1bi 328 . 2  |-  ( ps  <->  ( E. x  x  =  y  ->  ps )
)
82, 5, 73bitr4i 269 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  equsalh  2001  equsex  2002  equsexOLD  2003  dvelimf  2068  dvelimfOLD  2069  sb6x  2121  asymref2  5251  intirr  5252  fun11  5516  pm13.192  27587
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-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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