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Theorem ax9lem13 29774
Description: Lemma for ax9 1902. Similar to cbv3 1935 with distinct variables, without using sp 1728, ax9 1902, or ax10 1897. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax9lem13.a  |-  -.  A. w  -.  w  =  x
ax9lem13.c  |-  -.  A. x  -.  x  =  w
ax9lem13.d  |-  -.  A. x  -.  x  =  y
ax9lem13.1  |-  ( ph  ->  A. y ph )
ax9lem13.2  |-  ( ps 
->  A. x ps )
ax9lem13.3  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
ax9lem13  |-  ( A. x ph  ->  A. y ps )
Distinct variable groups:    x, y    x, w    ph, w    ps, w
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem ax9lem13
StepHypRef Expression
1 ax9lem13.1 . . 3  |-  ( ph  ->  A. y ph )
21alimi 1549 . 2  |-  ( A. x ph  ->  A. x A. y ph )
3 ax9lem13.d . . . . 5  |-  -.  A. x  -.  x  =  y
4 ax9lem13.a . . . . . . 7  |-  -.  A. w  -.  w  =  x
5 ax9lem13.c . . . . . . 7  |-  -.  A. x  -.  x  =  w
6 id 19 . . . . . . . 8  |-  ( ph  ->  ph )
76hbth 1542 . . . . . . . . 9  |-  ( (
ph  ->  ph )  ->  A. x
( ph  ->  ph )
)
84, 5ax9lem7 29768 . . . . . . . . . 10  |-  ( A. x ph  ->  A. x A. x ph )
98a1i 10 . . . . . . . . 9  |-  ( (
ph  ->  ph )  ->  ( A. x ph  ->  A. x A. x ph ) )
10 ax9lem13.2 . . . . . . . . . 10  |-  ( ps 
->  A. x ps )
1110a1i 10 . . . . . . . . 9  |-  ( (
ph  ->  ph )  ->  ( ps  ->  A. x ps )
)
124, 5, 7, 9, 11ax9lem9 29770 . . . . . . . 8  |-  ( (
ph  ->  ph )  ->  (
( A. x ph  ->  ps )  ->  A. x
( A. x ph  ->  ps ) ) )
136, 12ax-mp 8 . . . . . . 7  |-  ( ( A. x ph  ->  ps )  ->  A. x
( A. x ph  ->  ps ) )
144, 5, 13ax9lem8 29769 . . . . . 6  |-  ( -.  ( A. x ph  ->  ps )  ->  A. x  -.  ( A. x ph  ->  ps ) )
154, 5ax9lem3 29764 . . . . . . . 8  |-  ( A. x ph  ->  ph )
16 ax9lem13.3 . . . . . . . 8  |-  ( x  =  y  ->  ( ph  ->  ps ) )
1715, 16syl5 28 . . . . . . 7  |-  ( x  =  y  ->  ( A. x ph  ->  ps ) )
1817con3i 127 . . . . . 6  |-  ( -.  ( A. x ph  ->  ps )  ->  -.  x  =  y )
1914, 18alrimih 1555 . . . . 5  |-  ( -.  ( A. x ph  ->  ps )  ->  A. x  -.  x  =  y
)
203, 19mt3 171 . . . 4  |-  ( A. x ph  ->  ps )
2120alimi 1549 . . 3  |-  ( A. y A. x ph  ->  A. y ps )
2221a7s 1721 . 2  |-  ( A. x A. y ph  ->  A. y ps )
232, 22syl 15 1  |-  ( A. x ph  ->  A. y ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  ax9lem14  29775
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-8 1661  ax-6 1715  ax-7 1720  ax-11 1727
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