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Theorem cbvex 1686
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
cbvex.1  |-  F/ y
ph
cbvex.2  |-  F/ x ps
cbvex.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvex  |-  ( E. x ph  <->  E. y ps )

Proof of Theorem cbvex
StepHypRef Expression
1 cbvex.1 . . 3  |-  F/ y
ph
21nfri 1457 . 2  |-  ( ph  ->  A. y ph )
3 cbvex.2 . . 3  |-  F/ x ps
43nfri 1457 . 2  |-  ( ps 
->  A. x ps )
5 cbvex.3 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
62, 4, 5cbvexh 1685 1  |-  ( E. x ph  <->  E. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1394   E.wex 1426
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  sb8e  1785  cbvex2  1845  cbvmo  1988  mo23  1989  clelab  2212  cbvrexf  2585  issetf  2626  eqvincf  2740  rexab2  2779  cbvrexcsf  2989  abn0m  3306  rabn0m  3308  euabsn  3507  eluniab  3660  cbvopab1  3903  cbvopab2  3904  cbvopab1s  3905  intexabim  3980  iinexgm  3982  opeliunxp  4481  dfdmf  4617  dfrnf  4664  elrnmpt1  4674  cbvoprab1  5702  cbvoprab2  5703  opabex3d  5874  opabex3  5875  seq3f1olemp  9896  fsum2dlemstep  10791  bdsepnfALT  11426  strcollnfALT  11527
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