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Theorem ax11o 1947
Description: Derivation of set.mm's original ax-11o 2093 from ax-10 2092 and the shorter ax-11 1727 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2096 or ax-17 1606 (given all of the original and new versions of sp 1728 through ax-15 2095).

Another open problem is whether this theorem can be proved without relying on ax12o 1887.

Theorem ax11 2107 shows the reverse derivation of ax-11 1727 from ax-11o 2093.

Normally, ax11o 1947 should be used rather than ax-11o 2093, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Assertion
Ref Expression
ax11o  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )

Proof of Theorem ax11o
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax-11 1727 . 2  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
21ax11a2 1946 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 1530
This theorem is referenced by:  ax11b  1948  equs5  1949  ax11v  2049  a12study  29754  a12studyALT  29755  a12study3  29757
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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