Theorem ax11o 1875

Description: Derivation of set.mm's original ax-11o 1876 from the shorter ax-11 1559 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1867 or ax-17 1579.

Normally, ax11o 1875 should be used rather than ax-11o 1876, 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.

Step	Hyp	Ref	Expression
1		ax-11 1559	. 2 |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )
2	1	ax11a2 1874	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  ax11b  1879  equs5  1882