Theorem ax9 1949

Description: Theorem showing that ax-9 1654 follows from the weaker version ax9v 1655. (Even though this theorem depends on ax-9 1654, all references of ax-9 1654 are made via ax9v 1655. An earlier version stated ax9v 1655 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1654 so that all proofs can be traced back to ax9v 1655. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

Assertion

Ref	Expression
ax9	⊢ ¬ ∀x ¬ x = y

Proof of Theorem ax9

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sp 1747	. . 3 ⊢ (∀x ¬ x = y → ¬ x = y)
2		sp 1747	. . 3 ⊢ (∀x x = y → x = y)
3	1, 2	nsyl3 111	. 2 ⊢ (∀x x = y → ¬ ∀x ¬ x = y)
4		ax9v 1655	. . 3 ⊢ ¬ ∀v ¬ v = y
5		dveeq2 1940	. . . . . 6 ⊢ (¬ ∀x x = y → (v = y → ∀x v = y))
6		ax9v 1655	. . . . . . 7 ⊢ ¬ ∀x ¬ x = v
7		hba1 1786	. . . . . . . 8 ⊢ (∀x v = y → ∀x∀x v = y)
8		sp 1747	. . . . . . . . . 10 ⊢ (∀x v = y → v = y)
9		equequ2 1686	. . . . . . . . . 10 ⊢ (v = y → (x = v ↔ x = y))
10	8, 9	syl 15	. . . . . . . . 9 ⊢ (∀x v = y → (x = v ↔ x = y))
11	10	notbid 285	. . . . . . . 8 ⊢ (∀x v = y → (¬ x = v ↔ ¬ x = y))
12	7, 11	albidh 1590	. . . . . . 7 ⊢ (∀x v = y → (∀x ¬ x = v ↔ ∀x ¬ x = y))
13	6, 12	mtbii 293	. . . . . 6 ⊢ (∀x v = y → ¬ ∀x ¬ x = y)
14	5, 13	syl6com 31	. . . . 5 ⊢ (v = y → (¬ ∀x x = y → ¬ ∀x ¬ x = y))
15	14	con3i 127	. . . 4 ⊢ (¬ (¬ ∀x x = y → ¬ ∀x ¬ x = y) → ¬ v = y)
16	15	alrimiv 1631	. . 3 ⊢ (¬ (¬ ∀x x = y → ¬ ∀x ¬ x = y) → ∀v ¬ v = y)
17	4, 16	mt3 171	. 2 ⊢ (¬ ∀x x = y → ¬ ∀x ¬ x = y)
18	3, 17	pm2.61i 156	1 ⊢ ¬ ∀x ¬ x = y

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀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

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:  ax9o  1950  a9e  1951