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Theorem cbvrex 2499
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvral.1 yφ
cbvral.2 xψ
cbvral.3 (x = y → (φψ))
Assertion
Ref Expression
cbvrex (x A φy A ψ)
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbvrex
StepHypRef Expression
1 nfcv 2151 . 2 xA
2 nfcv 2151 . 2 yA
3 cbvral.1 . 2 yφ
4 cbvral.2 . 2 xψ
5 cbvral.3 . 2 (x = y → (φψ))
61, 2, 3, 4, 5cbvrexf 2497 1 (x A φy A ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wnf 1322  wrex 2276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-nf 1323  df-sb 1619  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rex 2281
This theorem is referenced by:  cbvrmo  2501  cbvrexv  2503  cbvrexsv  2514  cbviun  3657  rexxpf  4398  isarep1  4899  rexrnmpt  5223  elabrex  5310
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