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Theorem cbvexh 1713
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 1600 . . 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 140 . . . . . 6  |-  ( x  =  y  ->  ( ps 
<-> 
ph ) )
65equcoms 1669 . . . . 5  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
73, 6equsex 1691 . . . 4  |-  ( E. y ( y  =  x  /\  ps )  <->  ph )
8 simpr 109 . . . . 5  |-  ( ( y  =  x  /\  ps )  ->  ps )
98eximi 1564 . . . 4  |-  ( E. y ( y  =  x  /\  ps )  ->  E. y ps )
107, 9sylbir 134 . . 3  |-  ( ph  ->  E. y ps )
112, 10exlimih 1557 . 2  |-  ( E. x ph  ->  E. y ps )
123hbex 1600 . . 3  |-  ( E. x ph  ->  A. y E. x ph )
131, 4equsex 1691 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
14 simpr 109 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  ph )
1514eximi 1564 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ph )
1613, 15sylbir 134 . . 3  |-  ( ps 
->  E. x ph )
1712, 16exlimih 1557 . 2  |-  ( E. y ps  ->  E. x ph )
1811, 17impbii 125 1  |-  ( E. x ph  <->  E. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316   E.wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  cbvex  1714  sb8eh  1811  cbvexv  1872  euf  1982  mopick  2055
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