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Theorem ax11a2 1938
Description: Derive ax-11o 2085 from a hypothesis in the form of ax-11 1718. ax-10 2084 and ax-11 1718 are used by the proof, but not ax-10o 2083 or ax-11o 2085. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)
Hypothesis
Ref Expression
ax11a2.1  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
Assertion
Ref Expression
ax11a2  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Distinct variable groups:    x, z    y,
z    ph, z
Allowed substitution groups:    ph( x, y)

Proof of Theorem ax11a2
StepHypRef Expression
1 ax-17 1605 . . 3  |-  ( ph  ->  A. z ph )
2 ax11a2.1 . . 3  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
31, 2syl5 30 . 2  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
43ax11v2 1937 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   A.wal 1529
This theorem is referenced by:  ax11o  1939
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1531  df-nf 1534
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