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Theorem ax11 2107
Description: Rederivation of axiom ax-11 1727 from ax-11o 2093, ax-10o 2091, and other older axioms. The proof does not require ax-16 2096 or ax-17 1606. See theorem ax11o 1947 for the derivation of ax-11o 2093 from ax-11 1727.

An open problem is whether we can prove this using ax-10 2092 instead of ax-10o 2091.

This proof uses newer axioms ax-5 1547 and ax-9 1644, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2088 and ax-9o 2090. (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 228 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1-o 2106 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
3 ax-1 5 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ph ) )
43alimi 1549 . . . 4  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
52, 4syl6bir 220 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) )
65a1d 22 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) ) )
7 ax-4 2087 . . 3  |-  ( A. y ph  ->  ph )
8 ax-11o 2093 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
97, 8syl7 63 . 2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
106, 9pm2.61i 156 1  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  ax10o-o  2155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-7 1720  ax-4 2087  ax-5o 2088  ax-6o 2089  ax-10o 2091  ax-11o 2093  ax-12o 2094
This theorem depends on definitions:  df-bi 177
  Copyright terms: Public domain W3C validator