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Theorem ax11o 2033
Description: Derivation of set.mm's original ax-11o 2177 from ax-10 2176 and the shorter ax-11 1753 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2180 or ax-17 1623 (given all of the original and new versions of sp 1755 through ax-15 2179).

Another open problem is whether this theorem can be proved without relying on ax12o 1969.

Theorem ax11 2191 shows the reverse derivation of ax-11 1753 from ax-11o 2177.

Normally, ax11o 2033 should be used rather than ax-11o 2177, 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 1753 . 2  |-  ( x  =  z  ->  ( A. z ph  ->  A. x
( x  =  z  ->  ph ) ) )
21ax11a2 2032 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:  ax11b  2034  equs5  2035  ax11v  2131
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 1661  ax-8 1682  ax-6 1736  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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