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Theorem cbv3 1935
Description: Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2094. (Contributed by NM, 5-Aug-1993.)
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 . . . 4  |-  F/ y
ph
21a1i 10 . . 3  |-  (  T. 
->  F/ y ph )
3 cbv3.2 . . . 4  |-  F/ x ps
43a1i 10 . . 3  |-  (  T. 
->  F/ x ps )
5 cbv3.3 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
65a1i 10 . . 3  |-  (  T. 
->  ( x  =  y  ->  ( ph  ->  ps ) ) )
72, 4, 6cbv1 1932 . 2  |-  ( A. x A. y  T.  ->  ( A. x ph  ->  A. y ps ) )
8 tru 1312 . . 3  |-  T.
98ax-gen 1536 . 2  |-  A. y  T.
107, 9mpg 1538 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    T. wtru 1307   A.wal 1530   F/wnf 1534
This theorem is referenced by:  cbval  1937  ax16i  1999  ax16ALT2  2001  sb8  2045  mo  2178  ax10ext  27709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
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