Theorem ax12olem5 1931

Description: Lemma for ax12o 1934. See ax12olem6 1932 for derivation of ax12o 1934 from the conclusion. (Contributed by NM, 24-Dec-2015.)

Hypothesis

Ref	Expression
ax12olem5.1	⊢ (¬ x = y → (¬ ∀x ¬ y = z → ∀x y = z))

Assertion

Ref	Expression
ax12olem5	⊢ (¬ ∀x x = y → (y = z → ∀x y = z))

Proof of Theorem ax12olem5

Step	Hyp	Ref	Expression
1		exnal 1574	. 2 ⊢ (∃x ¬ x = y ↔ ¬ ∀x x = y)
2		19.8a 1756	. . 3 ⊢ (y = z → ∃x y = z)
3		hbe1 1731	. . . . 5 ⊢ (∃x y = z → ∀x∃x y = z)
4		hba1 1786	. . . . 5 ⊢ (∀x y = z → ∀x∀x y = z)
5	3, 4	hbim 1817	. . . 4 ⊢ ((∃x y = z → ∀x y = z) → ∀x(∃x y = z → ∀x y = z))
6		df-ex 1542	. . . . 5 ⊢ (∃x y = z ↔ ¬ ∀x ¬ y = z)
7		ax12olem5.1	. . . . 5 ⊢ (¬ x = y → (¬ ∀x ¬ y = z → ∀x y = z))
8	6, 7	syl5bi 208	. . . 4 ⊢ (¬ x = y → (∃x y = z → ∀x y = z))
9	5, 8	exlimih 1804	. . 3 ⊢ (∃x ¬ x = y → (∃x y = z → ∀x y = z))
10	2, 9	syl5 28	. 2 ⊢ (∃x ¬ x = y → (y = z → ∀x y = z))
11	1, 10	sylbir 204	1 ⊢ (¬ ∀x x = y → (y = z → ∀x y = z))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

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-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax12olem7  1933