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Theorem ax11indn 2271
Description: Induction step for constructing a substitution instance of ax-11o 2217 without using ax-11o 2217. Negation case. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax11indn.1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Assertion
Ref Expression
ax11indn  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )

Proof of Theorem ax11indn
StepHypRef Expression
1 19.8a 1762 . . 3  |-  ( ( x  =  y  /\  -.  ph )  ->  E. x
( x  =  y  /\  -.  ph )
)
2 exanali 1595 . . . 4  |-  ( E. x ( x  =  y  /\  -.  ph ) 
<->  -.  A. x ( x  =  y  ->  ph ) )
3 hbn1 1745 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
4 hbn1 1745 . . . . 5  |-  ( -. 
A. x ( x  =  y  ->  ph )  ->  A. x  -.  A. x ( x  =  y  ->  ph ) )
5 ax11indn.1 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
6 con3 128 . . . . . . 7  |-  ( (
ph  ->  A. x ( x  =  y  ->  ph )
)  ->  ( -.  A. x ( x  =  y  ->  ph )  ->  -.  ph ) )
75, 6syl6 31 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
A. x ( x  =  y  ->  ph )  ->  -.  ph ) ) )
87com23 74 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  -.  ph ) ) )
93, 4, 8alrimdh 1597 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  -.  ph )
) )
102, 9syl5bi 209 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  -.  ph )  ->  A. x ( x  =  y  ->  -.  ph ) ) )
111, 10syl5 30 . 2  |-  ( -. 
A. x  x  =  y  ->  ( (
x  =  y  /\  -.  ph )  ->  A. x
( x  =  y  ->  -.  ph ) ) )
1211exp3a 426 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550
This theorem is referenced by:  ax11indi  2272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
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