Theorem ax12olem1 1927

Description: Lemma for ax12o 1934. Similar to equvin 2001 but with a negated equality. (Contributed by NM, 24-Dec-2015.)

Assertion

Ref	Expression
ax12olem1	⊢ (∃w(y = w ∧ ¬ z = w) ↔ ¬ y = z)

Distinct variable groups:   y,w   z,w

Proof of Theorem ax12olem1

Step	Hyp	Ref	Expression
1		ax-8 1675	. . . . 5 ⊢ (y = w → (y = z → w = z))
2		equcomi 1679	. . . . 5 ⊢ (w = z → z = w)
3	1, 2	syl6 29	. . . 4 ⊢ (y = w → (y = z → z = w))
4	3	con3and 428	. . 3 ⊢ ((y = w ∧ ¬ z = w) → ¬ y = z)
5	4	exlimiv 1634	. 2 ⊢ (∃w(y = w ∧ ¬ z = w) → ¬ y = z)
6		ax-17 1616	. . 3 ⊢ (¬ y = z → ∀w ¬ y = z)
7		ax-8 1675	. . . . . . . 8 ⊢ (w = z → (w = y → z = y))
8		equcomi 1679	. . . . . . . 8 ⊢ (z = y → y = z)
9	7, 8	syl6 29	. . . . . . 7 ⊢ (w = z → (w = y → y = z))
10	9	equcoms 1681	. . . . . 6 ⊢ (z = w → (w = y → y = z))
11	10	com12 27	. . . . 5 ⊢ (w = y → (z = w → y = z))
12	11	con3d 125	. . . 4 ⊢ (w = y → (¬ y = z → ¬ z = w))
13		equcomi 1679	. . . 4 ⊢ (w = y → y = w)
14	12, 13	jctild 527	. . 3 ⊢ (w = y → (¬ y = z → (y = w ∧ ¬ z = w)))
15	6, 14	spimeh 1667	. 2 ⊢ (¬ y = z → ∃w(y = w ∧ ¬ z = w))
16	5, 15	impbii 180	1 ⊢ (∃w(y = w ∧ ¬ z = w) ↔ ¬ y = z)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∃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

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  ax12olem2  1928