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Theorem equsexh 1916
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
equsexh.1  |-  ( ps 
->  A. x ps )
equsexh.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsexh  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )

Proof of Theorem equsexh
StepHypRef Expression
1 equsexh.1 . . 3  |-  ( ps 
->  A. x ps )
21nfi 1541 . 2  |-  F/ x ps
3 equsexh.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
42, 3equsex 1915 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531
This theorem is referenced by:  cleljust  1967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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