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Theorem ax11a2 2050
Description: Derive ax-11o 2195 from a hypothesis in the form of ax-11 1757. ax-10 2194 and ax-11 1757 are used by the proof, but not ax-10o 2193 or ax-11o 2195. 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 hints:    ph( x, y)

Proof of Theorem ax11a2
StepHypRef Expression
1 ax-17 1623 . . 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 2048 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  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:  ax11o  2051
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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