Theorem ax12a2-o 34235

Description: Derive ax-c15 34174 from a hypothesis in the form of ax-12 2047, without using ax-12 2047 or ax-c15 34174. The hypothesis is weaker than ax-12 2047, with z both distinct from x and not occurring in ph. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 2047, if we also have ax-c11 34172, which this proof uses. As theorem ax12 2304 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 34173 instead of ax-c11 34172. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax12a2-o.1	|- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )

Assertion

Ref	Expression
ax12a2-o	|- ( -. 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 ax12a2-o

Step	Hyp	Ref	Expression
1		ax-5 1839	. . 3 |- ( ph -> A. z ph )
2		ax12a2-o.1	. . 3 |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )
3	1, 2	syl5 34	. 2 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
4	3	ax12v2-o 34234	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)