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Theorem equsex 1657
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Hypotheses
Ref Expression
equsex.1  |-  ( ps 
->  A. 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  |-  ( ps 
->  A. x ps )
2 equsex.2 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32biimpa 290 . . 3  |-  ( ( x  =  y  /\  ph )  ->  ps )
41, 3exlimih 1525 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  ps )
5 a9e 1627 . . 3  |-  E. x  x  =  y
6 idd 21 . . . . 5  |-  ( ps 
->  ( x  =  y  ->  x  =  y ) )
72biimprcd 158 . . . . 5  |-  ( ps 
->  ( x  =  y  ->  ph ) )
86, 7jcad 301 . . . 4  |-  ( ps 
->  ( x  =  y  ->  ( x  =  y  /\  ph )
) )
91, 8eximdh 1543 . . 3  |-  ( ps 
->  ( E. x  x  =  y  ->  E. x
( x  =  y  /\  ph ) ) )
105, 9mpi 15 . 2  |-  ( ps 
->  E. x ( x  =  y  /\  ph ) )
114, 10impbii 124 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   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
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cbvexh  1679  sb56  1807  cleljust  1855  sb10f  1913
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