Theorem ax12a2-o 2280

Description: Derive ax-c15 2220 from a hypothesis in the form of ax-12 1854, without using ax-12 1854 or ax-c15 2220. The hypothesis is even weaker than ax-12 1854, with both distinct from and not occurring in . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 1854, if we also hvae ax-c11 2218 which this proof uses . As theorem ax12 2234 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 2219 instead of ax-c11 2218. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax12a2-o.1

Assertion

Ref	Expression
ax12a2-o

Distinct variable groups:   ,   ,   ,

Proof of Theorem ax12a2-o

Step	Hyp	Ref	Expression
1		ax-5 1704	. . 3
2		ax12a2-o.1	. . 3
3	1, 2	syl5 32	. 2
4	3	ax12v2-o 2279	1

Syntax hints:  -.wn 3  ->wi 4  A.wal 1393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-c5 2214  ax-c4 2215  ax-c7 2216  ax-c11 2218  ax-c9 2221

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617