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Theorem equsex 1915
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
equsex.1  |-  F/ 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 exnal 1564 . 2  |-  ( E. x  -.  ( x  =  y  ->  -.  ph )  <->  -.  A. x
( x  =  y  ->  -.  ph ) )
2 df-an 360 . . 3  |-  ( ( x  =  y  /\  ph )  <->  -.  ( x  =  y  ->  -.  ph ) )
32exbii 1572 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  E. x  -.  ( x  =  y  ->  -.  ph ) )
4 equsex.1 . . . . 5  |-  F/ x ps
54nfn 1777 . . . 4  |-  F/ x  -.  ps
6 equsex.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
76notbid 285 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
85, 7equsal 1913 . . 3  |-  ( A. x ( x  =  y  ->  -.  ph )  <->  -. 
ps )
98con2bii 322 . 2  |-  ( ps  <->  -. 
A. x ( x  =  y  ->  -.  ph ) )
101, 3, 93bitr4i 268 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531   F/wnf 1534
This theorem is referenced by:  equsexh  1916  cleljustALT  1968  sb56  2050  sb10f  2074  axsep  4156
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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