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Theorem ax12indi 32440
Description: Induction step for constructing a substitution instance of ax-c15 32386 without using ax-c15 32386. Implication case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax12indn.1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
ax12indi.2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ps 
->  A. x ( x  =  y  ->  ps ) ) ) )
Assertion
Ref Expression
ax12indi  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( (
ph  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) ) )

Proof of Theorem ax12indi
StepHypRef Expression
1 ax12indn.1 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
21ax12indn 32439 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )
32imp 431 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( -.  ph  ->  A. x ( x  =  y  ->  -.  ph )
) )
4 pm2.21 112 . . . . . 6  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
54imim2i 16 . . . . 5  |-  ( ( x  =  y  ->  -.  ph )  ->  (
x  =  y  -> 
( ph  ->  ps )
) )
65alimi 1681 . . . 4  |-  ( A. x ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )
73, 6syl6 35 . . 3  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( -.  ph  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) )
8 ax12indi.2 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ps 
->  A. x ( x  =  y  ->  ps ) ) ) )
98imp 431 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ps  ->  A. x
( x  =  y  ->  ps ) ) )
10 ax-1 6 . . . . . 6  |-  ( ps 
->  ( ph  ->  ps ) )
1110imim2i 16 . . . . 5  |-  ( ( x  =  y  ->  ps )  ->  ( x  =  y  ->  ( ph  ->  ps ) ) )
1211alimi 1681 . . . 4  |-  ( A. x ( x  =  y  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )
139, 12syl6 35 . . 3  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ps  ->  A. x
( x  =  y  ->  ( ph  ->  ps ) ) ) )
147, 13jad 166 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ( ph  ->  ps )  ->  A. x
( x  =  y  ->  ( ph  ->  ps ) ) ) )
1514ex 436 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( (
ph  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371   A.wal 1436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-12 1906
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661
This theorem is referenced by: (None)
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