Theorem ax11a2-o 2202

Description: Derive ax-11o 2141 from a hypothesis in the form of ax-11 1746, without using ax-11 1746 or ax-11o 2141. The hypothesis is even weaker than ax-11 1746, with z both distinct from x and not occurring in φ. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11 1746, if we also hvae ax-10o 2139 which this proof uses . As Theorem ax11 2155 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2140 instead of ax-10o 2139. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax11a2-o.1	⊢ (x = z → (∀zφ → ∀x(x = z → φ)))

Assertion

Ref	Expression
ax11a2-o	⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

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

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11a2-o

Step	Hyp	Ref	Expression
1		ax-17 1616	. . 3 ⊢ (φ → ∀zφ)
2		ax11a2-o.1	. . 3 ⊢ (x = z → (∀zφ → ∀x(x = z → φ)))
3	1, 2	syl5 28	. 2 ⊢ (x = z → (φ → ∀x(x = z → φ)))
4	3	ax11v2-o 2201	1 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142

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: (None)