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Theorem equsex 1689
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 292 . . 3  |-  ( ( x  =  y  /\  ph )  ->  ps )
41, 3exlimih 1555 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  ps )
5 a9e 1657 . . 3  |-  E. x  x  =  y
6 idd 21 . . . . 5  |-  ( ps 
->  ( x  =  y  ->  x  =  y ) )
72biimprcd 159 . . . . 5  |-  ( ps 
->  ( x  =  y  ->  ph ) )
86, 7jcad 303 . . . 4  |-  ( ps 
->  ( x  =  y  ->  ( x  =  y  /\  ph )
) )
91, 8eximdh 1573 . . 3  |-  ( ps 
->  ( E. x  x  =  y  ->  E. x
( x  =  y  /\  ph ) ) )
105, 9mpi 15 . 2  |-  ( ps 
->  E. x ( x  =  y  /\  ph ) )
114, 10impbii 125 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    = wceq 1314   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
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  cbvexh  1711  sb56  1839  cleljust  1888  sb10f  1946
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