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Theorem equsexh 2004
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 1560 . 2  |-  F/ x ps
3 equsexh.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
42, 3equsex 2002 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550
This theorem is referenced by:  cleljust  2097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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