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Theorem equsex 2129
Description: An equivalence related to implicit substitution. See equsexv 2065 for a version with a dv condition which does not require ax-13 2090. See equsexALT 2130 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)
Hypotheses
Ref Expression
equsex.nf  |-  F/ x ps
equsex.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsex  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )

Proof of Theorem equsex
StepHypRef Expression
1 equsex.nf . . 3  |-  F/ x ps
2 equsex.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32biimpa 487 . . 3  |-  ( ( x  =  y  /\  ph )  ->  ps )
41, 3exlimi 1994 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  ps )
51, 2equsal 2127 . . 3  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
6 equs4 2126 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
75, 6sylbir 217 . 2  |-  ( ps 
->  E. x ( x  =  y  /\  ph ) )
84, 7impbii 191 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1441   E.wex 1662   F/wnf 1666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-12 1932  ax-13 2090
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1663  df-nf 1667
This theorem is referenced by:  equsexh  2131  sb5rf  2250
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