Theorem ax11a2 1993

Description: Derive ax-11o 2141 from a hypothesis in the form of ax-11 1746. ax-10 2140 and ax-11 1746 are used by the proof, but not ax-10o 2139 or ax-11o 2141. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)

Hypothesis

Ref	Expression
ax11a2.1	|- (x = z -> (A.zph -> A.x(x = z -> ph)))

Assertion

Ref	Expression
ax11a2	|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Distinct variable groups:   x,z   y,z   ph,z

Allowed substitution hints:   ph(x,y)

Proof of Theorem ax11a2

Step	Hyp	Ref	Expression
1		ax-17 1616	. . 3 |- (ph -> A.zph)
2		ax11a2.1	. . 3 |- (x = z -> (A.zph -> A.x(x = z -> ph)))
3	1, 2	syl5 28	. 2 |- (x = z -> (ph -> A.x(x = z -> ph)))
4	3	ax11v2 1992	1 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax11o  1994