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Theorem ax12indn 2275
Description: Induction step for constructing a substitution instance of ax-c15 2222 without using ax-c15 2222. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12indn.1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Assertion
Ref Expression
ax12indn  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )

Proof of Theorem ax12indn
StepHypRef Expression
1 19.8a 1862 . . 3  |-  ( ( x  =  y  /\  -.  ph )  ->  E. x
( x  =  y  /\  -.  ph )
)
2 exanali 1675 . . . 4  |-  ( E. x ( x  =  y  /\  -.  ph ) 
<->  -.  A. x ( x  =  y  ->  ph ) )
3 hbn1 1843 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
4 hbn1 1843 . . . . 5  |-  ( -. 
A. x ( x  =  y  ->  ph )  ->  A. x  -.  A. x ( x  =  y  ->  ph ) )
5 ax12indn.1 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
6 con3 134 . . . . . . 7  |-  ( (
ph  ->  A. x ( x  =  y  ->  ph )
)  ->  ( -.  A. x ( x  =  y  ->  ph )  ->  -.  ph ) )
75, 6syl6 33 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
A. x ( x  =  y  ->  ph )  ->  -.  ph ) ) )
87com23 78 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  -.  ph ) ) )
93, 4, 8alrimdh 1677 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  -.  ph )
) )
102, 9syl5bi 217 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  -.  ph )  ->  A. x ( x  =  y  ->  -.  ph ) ) )
111, 10syl5 32 . 2  |-  ( -. 
A. x  x  =  y  ->  ( (
x  =  y  /\  -.  ph )  ->  A. x
( x  =  y  ->  -.  ph ) ) )
1211expd 434 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1396   E.wex 1617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618
This theorem is referenced by:  ax12indi  2276
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