Theorem ax9from9o 2148

Description: Rederivation of Axiom ax-9 1654 from ax-9o 2138 and other older axioms. See ax9o 1950 for the derivation of ax-9o 2138 from ax-9 1654. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax9from9o	⊢ ¬ ∀x ¬ x = y

Proof of Theorem ax9from9o

Step	Hyp	Ref	Expression
1		ax-9o 2138	. 2 ⊢ (∀x(x = y → ∀x ¬ ∀x ¬ x = y) → ¬ ∀x ¬ x = y)
2		ax-6o 2137	. . 3 ⊢ (¬ ∀x ¬ ∀x ¬ x = y → ¬ x = y)
3	2	con4i 122	. 2 ⊢ (x = y → ∀x ¬ ∀x ¬ x = y)
4	1, 3	mpg 1548	1 ⊢ ¬ ∀x ¬ x = y

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-6o 2137  ax-9o 2138

This theorem is referenced by:  equidqe  2173