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Theorem cbv3hv 1737
Description: Lemma for ax10 1884. Similar to cbv3h 1923. Requires distinct variables but avoids ax-12 1866. (Contributed by NM, 25-Jul-2015.)
Hypotheses
Ref Expression
cbv3hv.1  |-  ( ph  ->  A. y ph )
cbv3hv.2  |-  ( ps 
->  A. x ps )
cbv3hv.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
cbv3hv  |-  ( A. x ph  ->  A. y ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbv3hv
StepHypRef Expression
1 cbv3hv.1 . . 3  |-  ( ph  ->  A. y ph )
21alimi 1546 . 2  |-  ( A. x ph  ->  A. x A. y ph )
3 ax9v 1636 . . . . 5  |-  -.  A. x  -.  x  =  y
4 hba1 1719 . . . . . . . 8  |-  ( A. x ph  ->  A. x A. x ph )
5 cbv3hv.2 . . . . . . . 8  |-  ( ps 
->  A. x ps )
64, 5hbim 1725 . . . . . . 7  |-  ( ( A. x ph  ->  ps )  ->  A. x
( A. x ph  ->  ps ) )
76hbn 1720 . . . . . 6  |-  ( -.  ( A. x ph  ->  ps )  ->  A. x  -.  ( A. x ph  ->  ps ) )
8 sp 1716 . . . . . . . 8  |-  ( A. x ph  ->  ph )
9 cbv3hv.3 . . . . . . . 8  |-  ( x  =  y  ->  ( ph  ->  ps ) )
108, 9syl5 28 . . . . . . 7  |-  ( x  =  y  ->  ( A. x ph  ->  ps ) )
1110con3i 127 . . . . . 6  |-  ( -.  ( A. x ph  ->  ps )  ->  -.  x  =  y )
127, 11alrimih 1552 . . . . 5  |-  ( -.  ( A. x ph  ->  ps )  ->  A. x  -.  x  =  y
)
133, 12mt3 171 . . . 4  |-  ( A. x ph  ->  ps )
1413alimi 1546 . . 3  |-  ( A. y A. x ph  ->  A. y ps )
1514a7s 1709 . 2  |-  ( A. x A. y ph  ->  A. y ps )
162, 15syl 15 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
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
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