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Theorem cbv3h 1923
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cbv3h.1  |-  ( ph  ->  A. y ph )
cbv3h.2  |-  ( ps 
->  A. x ps )
cbv3h.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbv3h  |-  ( A. x ph  ->  A. y ps )

Proof of Theorem cbv3h
StepHypRef Expression
1 cbv3h.1 . . . 4  |-  ( ph  ->  A. y ph )
21a1i 10 . . 3  |-  ( y  =  y  ->  ( ph  ->  A. y ph )
)
3 cbv3h.2 . . . 4  |-  ( ps 
->  A. x ps )
43a1i 10 . . 3  |-  ( y  =  y  ->  ( ps  ->  A. x ps )
)
5 cbv3h.3 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
65a1i 10 . . 3  |-  ( y  =  y  ->  (
x  =  y  -> 
( ph  ->  ps )
) )
72, 4, 6cbv1h 1918 . 2  |-  ( A. x A. y  y  =  y  ->  ( A. x ph  ->  A. y ps ) )
8 stdpc6 1650 . 2  |-  A. y 
y  =  y
97, 8mpg 1535 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem is referenced by:  cleqh  2380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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