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Theorem equs45f 1724
Description: Two ways of expressing substitution when  y is not free in  ph. (Contributed by NM, 25-Apr-2008.)
Hypothesis
Ref Expression
equs45f.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
equs45f  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )

Proof of Theorem equs45f
StepHypRef Expression
1 equs45f.1 . . . . 5  |-  ( ph  ->  A. y ph )
21anim2i 334 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  A. y ph ) )
32eximi 1532 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  A. y ph ) )
4 equs5a 1716 . . 3  |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x
( x  =  y  ->  ph ) )
53, 4syl 14 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)
6 equs4 1654 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
75, 6impbii 124 1  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   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-11 1438  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  sb5f  1726
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