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Theorem ax11 1942
Description: Rederivation of axiom ax-11 1624 from the orginal version, ax-11o 1941, and ax-10o 1836. The proof does not require ax-11 1624, ax-16 1927, or ax-17 1628. See theorem ax11o 1940 for the derivation of ax-11o 1941 from ax-11 1624.

An open problem is whether we can prove this using ax-10 1678 instead of ax-10o 1836.

This theorem should not be referenced in any proof. Instead, use ax-11 1624 above so that uses of ax-11 1624 can be more easily identified. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax11  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )

Proof of Theorem ax11
StepHypRef Expression
1 biidd 230 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1-o 1857 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
3 ax-1 7 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ph ) )
43alimi 1546 . . . 4  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
52, 4syl6bir 222 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) )
65a1d 24 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) ) )
7 ax-4 1692 . . 3  |-  ( A. y ph  ->  ph )
8 ax-11o 1941 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
97, 8syl7 65 . 2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
106, 9pm2.61i 158 1  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   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-7 1535  ax-gen 1536  ax-8 1623  ax-12o 1664  ax-9 1684  ax-4 1692  ax-10o 1836  ax-11o 1941
This theorem depends on definitions:  df-bi 179  df-nf 1540
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