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Theorem ax11o 1939
Description: Derivation of set.mm's original ax-11o 2085 from ax-10 2084 and the shorter ax-11 1718 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2088 or ax-17 1605 (given all of the original and new versions of sp 1719 through ax-15 2087).

Another open problem is whether this theorem can be proved without relying on ax12o 1879.

Theorem ax11 2098 shows the reverse derivation of ax-11 1718 from ax-11o 2085.

Normally, ax11o 1939 should be used rather than ax-11o 2085, 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 )
) ) )
Dummy variable  z is distinct from all other variables.

Proof of Theorem ax11o
StepHypRef Expression
1 ax-11 1718 . 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 1529
This theorem is referenced by:  ax11b  1940  equs5  1941  ax11v  2038  a12study  28401  a12studyALT  28402  a12study3  28404
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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