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Theorem cbvralw 2712
Description: Rule used to change bound variables, using implicit substitution. Version of cbvral 2714 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1518 and ax-bndl 1520 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvralw.1  |-  F/ y
ph
cbvralw.2  |-  F/ x ps
cbvralw.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvralw  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvralw
StepHypRef Expression
1 nfcv 2332 . 2  |-  F/_ x A
2 nfcv 2332 . 2  |-  F/_ y A
3 cbvralw.1 . 2  |-  F/ y
ph
4 cbvralw.2 . 2  |-  F/ x ps
5 cbvralw.3 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
61, 2, 3, 4, 5cbvralfw 2708 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   F/wnf 1471   A.wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473
This theorem is referenced by:  pcmptdvds  12361  lgseisenlem2  14848
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