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Theorem cbvexd 1949
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1956. (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 1760 . . . 4  |-  ( ph  ->  F/ y  -.  ps )
4 cbvald.3 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
5 notbi 286 . . . . 5  |-  ( ( ps  <->  ch )  <->  ( -.  ps 
<->  -.  ch ) )
64, 5syl6ib 217 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ( -.  ps  <->  -. 
ch ) ) )
71, 3, 6cbvald 1948 . . 3  |-  ( ph  ->  ( A. x  -.  ps 
<-> 
A. y  -.  ch ) )
87notbid 285 . 2  |-  ( ph  ->  ( -.  A. x  -.  ps  <->  -.  A. y  -.  ch ) )
9 df-ex 1529 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
10 df-ex 1529 . 2  |-  ( E. y ch  <->  -.  A. y  -.  ch )
118, 9, 103bitr4g 279 1  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528   F/wnf 1531
This theorem is referenced by:  cbvexdva  1951  vtoclgft  2834  dfid3  4310  isinf  7076  axrepndlem2  8215  axunnd  8218  axpowndlem2  8220  axpownd  8223  axregndlem2  8225  axinfndlem1  8227  axacndlem4  8232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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