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Theorem cbv3 2042
Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 31914. (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 1975 . 2  |-  F/ y A. x ph
3 cbv3.2 . . 3  |-  F/ x ps
4 cbv3.3 . . 3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
53, 4spim 2033 . 2  |-  ( A. x ph  ->  ps )
62, 5alrimi 1901 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1403   F/wnf 1637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638
This theorem is referenced by:  cbv3h  2043  cbv1  2044  cbval  2048  axc16i  2090  mo3OLD  2279
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