Theorem ax9a 808

Description: This theorem is a re-derivation of ax-9 799 from ax9 807. This shows that ax-9 799 and ax9 807 are interchangeable in the presence of the other axioms. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint). Use it instead of ax-9 799 so we interchange ax-9 799 and ax9 807 as our axiom.

Assertion

Ref	Expression
ax9a	⊢ ¬ ∀x ¬ x = y

Proof of Theorem ax9a

Step	Hyp	Ref	Expression
1		ax9 807	. 2 ⊢ (∀x(x = y → ∀x ¬ ∀x ¬ x = y) → ¬ ∀x ¬ x = y)
2		ax-6 675	. . 3 ⊢ (¬ ∀x ¬ ∀x ¬ x = y → ¬ x = y)
3	2	a3i 69	. 2 ⊢ (x = y → ∀x ¬ ∀x ¬ x = y)
4	1, 3	mpg 684	1 ⊢ ¬ ∀x ¬ x = y

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672   = weq 797

This theorem is referenced by:  a9e 809  a16g 933

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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