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Theorem ax11o 1940
Description: Derivation of set.mm's original ax-11o 1941 from ax-10 1678 and the shorter ax-11 1624 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1927 or ax-17 1628 (given all of the original and new versions of ax-4 1692 through ax-15 2105).

Another open problem is whether this theorem can be proved without relying on ax-12o 1664.

Theorem ax11 1942 shows the reverse derivation of ax-11 1624 from ax-11o 1941.

Normally, ax11o 1940 should be used rather than ax-11o 1941, 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
StepHypRef Expression
1 ax-11 1624 . 2  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
21ax11a2 1938 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 1532
This theorem is referenced by:  equs5  1944  ax11v  1991
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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