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Theorem cbv3 1971
Description: Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2219. (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 1864 . 2  |-  F/ y A. x ph
3 cbv3.2 . . 3  |-  F/ x ps
4 cbv3.3 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
53, 4spim 1957 . 2  |-  ( A. x ph  ->  ps )
62, 5alrimi 1781 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   F/wnf 1553
This theorem is referenced by:  cbv3h  1972  cbv1  1973  cbval  1982  ax16i  2051  ax16ALT2  2155  mo  2303  ax10ext  27583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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