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Theorem cbvexd 2045
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2052. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
cbvald.1  |-  F/ y
ph
cbvald.2  |-  ( ph  ->  F/ y ps )
cbvald.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbvexd  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Distinct variable groups:    ph, x    ch, x
Allowed substitution hints:    ph( y)    ps( x, y)    ch( y)

Proof of Theorem cbvexd
StepHypRef Expression
1 cbvald.1 . . . 4  |-  F/ y
ph
2 cbvald.2 . . . . 5  |-  ( ph  ->  F/ y ps )
32nfnd 1799 . . . 4  |-  ( ph  ->  F/ y  -.  ps )
4 cbvald.3 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
5 notbi 287 . . . . 5  |-  ( ( ps  <->  ch )  <->  ( -.  ps 
<->  -.  ch ) )
64, 5syl6ib 218 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ( -.  ps  <->  -. 
ch ) ) )
71, 3, 6cbvald 2044 . . 3  |-  ( ph  ->  ( A. x  -.  ps 
<-> 
A. y  -.  ch ) )
87notbid 286 . 2  |-  ( ph  ->  ( -.  A. x  -.  ps  <->  -.  A. y  -.  ch ) )
9 df-ex 1548 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
10 df-ex 1548 . 2  |-  ( E. y ch  <->  -.  A. y  -.  ch )
118, 9, 103bitr4g 280 1  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547   F/wnf 1550
This theorem is referenced by:  cbvexdva  2047  vtoclgft  2947  dfid3  4442  axrepndlem2  8403  axunnd  8406  axpowndlem2  8408  axpownd  8411  axregndlem2  8413  axinfndlem1  8415  axacndlem4  8420
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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