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Theorem ax11 2189
Description: Rederivation of axiom ax-11 1753 from ax-11o 2175, ax-10o 2173, and other older axioms. The proof does not require ax-16 2178 or ax-17 1623. See theorem ax11o 2028 for the derivation of ax-11o 2175 from ax-11 1753.

An open problem is whether we can prove this using ax-10 2174 instead of ax-10o 2173.

This proof uses newer axioms ax-5 1563 and ax-9 1661, 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 2170 and ax-9o 2172. (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 229 . . . . 5  |-  ( A. x  x  =  y  ->  ( ph  <->  ph ) )
21dral1-o 2188 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x ph  <->  A. y ph ) )
3 ax-1 5 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ph ) )
43alimi 1565 . . . 4  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
52, 4syl6bir 221 . . 3  |-  ( A. x  x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) )
65a1d 23 . 2  |-  ( A. x  x  =  y  ->  ( x  =  y  ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
) ) )
7 ax-4 2169 . . 3  |-  ( A. y ph  ->  ph )
8 ax-11o 2175 . . 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 3    -> wi 4   A.wal 1546
This theorem is referenced by:  ax10o-o  2237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-7 1741  ax-4 2169  ax-5o 2170  ax-6o 2171  ax-10o 2173  ax-11o 2175  ax-12o 2176
This theorem depends on definitions:  df-bi 178  df-ex 1548
  Copyright terms: Public domain W3C validator