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Theorem ax11a2-o 2141
Description: Derive ax-11o 2080 from a hypothesis in the form of ax-11 1715, without using ax-11 1715 or ax-11o 2080. The hypothesis is even weaker than ax-11 1715, with  z both distinct from  x and not occurring in  ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax-11 1715, if we also hvae ax-10o 2078 which this proof uses . As theorem ax11 2094 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2079 instead of ax-10o 2078. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax11a2-o.1  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
Assertion
Ref Expression
ax11a2-o  |-  ( -. 
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-o
StepHypRef Expression
1 ax-17 1603 . . 3  |-  ( ph  ->  A. z ph )
2 ax11a2-o.1 . . 3  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
31, 2syl5 28 . 2  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
43ax11v2-o 2140 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 1527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-4 2074  ax-5o 2075  ax-6o 2076  ax-10o 2078  ax-12o 2081
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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