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Theorem cbvexh 1680
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
cbvexh.1  |-  ( ph  ->  A. y ph )
cbvexh.2  |-  ( ps 
->  A. x ps )
cbvexh.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvexh  |-  ( E. x ph  <->  E. y ps )

Proof of Theorem cbvexh
StepHypRef Expression
1 cbvexh.2 . . . 4  |-  ( ps 
->  A. x ps )
21hbex 1568 . . 3  |-  ( E. y ps  ->  A. x E. y ps )
3 cbvexh.1 . . . . 5  |-  ( ph  ->  A. y ph )
4 cbvexh.3 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
54bicomd 139 . . . . . 6  |-  ( x  =  y  ->  ( ps 
<-> 
ph ) )
65equcoms 1636 . . . . 5  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
73, 6equsex 1658 . . . 4  |-  ( E. y ( y  =  x  /\  ps )  <->  ph )
8 simpr 108 . . . . 5  |-  ( ( y  =  x  /\  ps )  ->  ps )
98eximi 1532 . . . 4  |-  ( E. y ( y  =  x  /\  ps )  ->  E. y ps )
107, 9sylbir 133 . . 3  |-  ( ph  ->  E. y ps )
112, 10exlimih 1525 . 2  |-  ( E. x ph  ->  E. y ps )
123hbex 1568 . . 3  |-  ( E. x ph  ->  A. y E. x ph )
131, 4equsex 1658 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
14 simpr 108 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  ph )
1514eximi 1532 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ph )
1613, 15sylbir 133 . . 3  |-  ( ps 
->  E. x ph )
1712, 16exlimih 1525 . 2  |-  ( E. y ps  ->  E. x ph )
1811, 17impbii 124 1  |-  ( E. x ph  <->  E. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cbvex  1681  sb8eh  1778  cbvexv  1838  euf  1948  mopick  2021
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