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Theorem cbvexd 1818
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1909. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
Hypotheses
Ref Expression
cbvexd.1  |-  F/ y
ph
cbvexd.2  |-  ( ph  ->  F/ y ps )
cbvexd.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 cbvexd.1 . . 3  |-  F/ y
ph
21nfri 1428 . 2  |-  ( ph  ->  A. y ph )
3 cbvexd.2 . . 3  |-  ( ph  ->  F/ y ps )
43nfrd 1429 . 2  |-  ( ph  ->  ( ps  ->  A. y ps ) )
5 cbvexd.3 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
62, 4, 5cbvexdh 1817 1  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   F/wnf 1365   E.wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  cbvexdva  1820  vtoclgft  2621  bdsepnft  10394  strcollnft  10496
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