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Theorem equsex 2003
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)
Hypotheses
Ref Expression
equsex.1  |-  F/ x ps
equsex.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsex  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )

Proof of Theorem equsex
StepHypRef Expression
1 equsex.1 . . 3  |-  F/ x ps
2 equsex.2 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32biimpa 472 . . 3  |-  ( ( x  =  y  /\  ph )  ->  ps )
41, 3exlimi 1822 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  ps )
51, 2equsal 2000 . . 3  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
6 equs4 1998 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
75, 6sylbir 206 . 2  |-  ( ps 
->  E. x ( x  =  y  /\  ph ) )
84, 7impbii 182 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551   F/wnf 1554
This theorem is referenced by:  equsexh  2005  cleljustALT  2106  sb56  2176  sb10f  2201  axsep  4331  dprd2d2  15604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555
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