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Theorem cbv3 1999
Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 2205. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3.1  |-  F/ y
ph
cbv3.2  |-  F/ x ps
cbv3.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbv3  |-  ( A. x ph  ->  A. y ps )

Proof of Theorem cbv3
StepHypRef Expression
1 cbv3.1 . . 3  |-  F/ y
ph
21nfal 1931 . 2  |-  F/ y A. x ph
3 cbv3.2 . . 3  |-  F/ x ps
4 cbv3.3 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
53, 4spim 1990 . 2  |-  ( A. x ph  ->  ps )
62, 5alrimi 1861 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1379   F/wnf 1601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1598  df-nf 1602
This theorem is referenced by:  cbv3h  2000  cbv1  2001  cbval  2005  axc16i  2048  mo3OLD  2308
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