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Theorem ax11indi 2112
Description: Induction step for constructing a substitution instance of ax-11o 1941 without using ax-11o 1941. Implication case. (Contributed by NM, 21-Jan-2007.)
Hypotheses
Ref Expression
ax11indn.1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
ax11indi.2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ps 
->  A. x ( x  =  y  ->  ps ) ) ) )
Assertion
Ref Expression
ax11indi  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( (
ph  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) ) )

Proof of Theorem ax11indi
StepHypRef Expression
1 ax11indn.1 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
21ax11indn 2111 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( -. 
ph  ->  A. x ( x  =  y  ->  -.  ph ) ) ) )
32imp 420 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( -.  ph  ->  A. x ( x  =  y  ->  -.  ph )
) )
4 pm2.21 102 . . . . . 6  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
54imim2i 15 . . . . 5  |-  ( ( x  =  y  ->  -.  ph )  ->  (
x  =  y  -> 
( ph  ->  ps )
) )
65alimi 1546 . . . 4  |-  ( A. x ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )
73, 6syl6 31 . . 3  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( -.  ph  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) )
8 ax11indi.2 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ps 
->  A. x ( x  =  y  ->  ps ) ) ) )
98imp 420 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ps  ->  A. x
( x  =  y  ->  ps ) ) )
10 ax-1 7 . . . . . 6  |-  ( ps 
->  ( ph  ->  ps ) )
1110imim2i 15 . . . . 5  |-  ( ( x  =  y  ->  ps )  ->  ( x  =  y  ->  ( ph  ->  ps ) ) )
1211alimi 1546 . . . 4  |-  ( A. x ( x  =  y  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )
139, 12syl6 31 . . 3  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ps  ->  A. x
( x  =  y  ->  ( ph  ->  ps ) ) ) )
147, 13jad 156 . 2  |-  ( ( -.  A. x  x  =  y  /\  x  =  y )  -> 
( ( ph  ->  ps )  ->  A. x
( x  =  y  ->  ( ph  ->  ps ) ) ) )
1514ex 425 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( (
ph  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   A.wal 1532
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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