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Theorem equsexNEW7 29196
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
Hypotheses
Ref Expression
equsex.1NEW  |-  F/ x ps
equsex.2NEW  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsexNEW7  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )

Proof of Theorem equsexNEW7
StepHypRef Expression
1 exnal 1580 . 2  |-  ( E. x  -.  ( x  =  y  ->  -.  ph )  <->  -.  A. x
( x  =  y  ->  -.  ph ) )
2 df-an 361 . . 3  |-  ( ( x  =  y  /\  ph )  <->  -.  ( x  =  y  ->  -.  ph ) )
32exbii 1589 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  E. x  -.  ( x  =  y  ->  -.  ph ) )
4 equsex.1NEW . . . . 5  |-  F/ x ps
54nfn 1807 . . . 4  |-  F/ x  -.  ps
6 equsex.2NEW . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
76notbid 286 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
85, 7equsalNEW7 29193 . . 3  |-  ( A. x ( x  =  y  ->  -.  ph )  <->  -. 
ps )
98con2bii 323 . 2  |-  ( ps  <->  -. 
A. x ( x  =  y  ->  -.  ph ) )
101, 3, 93bitr4i 269 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547   F/wnf 1550
This theorem is referenced by:  equsexhNEW7  29197  sb56NEW7  29297  sb10fOLD7  29451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-11 1757  ax-12 1946  ax-7v 29148
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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