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Theorem ax11indn 2111
Description: Induction step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. Negation case. (Contributed by NM, 21-Jan-2007.)
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 1758 . . 3  |-  ( ( x  =  y  /\  -.  ph )  ->  E. x
( x  =  y  /\  -.  ph )
)
2 exanali 1583 . . . 4  |-  ( E. x ( x  =  y  /\  -.  ph ) 
<->  -.  A. x ( x  =  y  ->  ph ) )
3 hbn1 1564 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
4 hbn1 1564 . . . . 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 1585 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x ( x  =  y  ->  ph )  ->  A. x ( x  =  y  ->  -.  ph )
) )
102, 9syl5bi 210 . . 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 427 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   A.wal 1532   E.wex 1537
This theorem is referenced by:  ax11indi  2112
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538
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