Theorem ax12olem6 1932

Description: Lemma for ax12o 1934. Derivation of ax12o 1934 from the hypotheses, without using ax12o 1934. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.)

Hypotheses

Ref	Expression
ax12olem6.1	⊢ (¬ ∀x x = z → (z = w → ∀x z = w))
ax12olem6.2	⊢ (¬ ∀x x = y → (y = w → ∀x y = w))

Assertion

Ref	Expression
ax12olem6	⊢ (¬ ∀x x = y → (¬ ∀x x = z → (y = z → ∀x y = z)))

Distinct variable groups:   x,w   y,w   z,w

Proof of Theorem ax12olem6

Step	Hyp	Ref	Expression
1		hbn1 1730	. . . . 5 ⊢ (¬ ∀x x = z → ∀x ¬ ∀x x = z)
2		ax12olem6.1	. . . . 5 ⊢ (¬ ∀x x = z → (z = w → ∀x z = w))
3	1, 2	hbim1 1810	. . . 4 ⊢ ((¬ ∀x x = z → z = w) → ∀x(¬ ∀x x = z → z = w))
4		ax-17 1616	. . . 4 ⊢ ((¬ ∀x x = z → y = z) → ∀w(¬ ∀x x = z → y = z))
5		equcom 1680	. . . . . 6 ⊢ (z = w ↔ w = z)
6		equequ1 1684	. . . . . 6 ⊢ (w = y → (w = z ↔ y = z))
7	5, 6	syl5bb 248	. . . . 5 ⊢ (w = y → (z = w ↔ y = z))
8	7	imbi2d 307	. . . 4 ⊢ (w = y → ((¬ ∀x x = z → z = w) ↔ (¬ ∀x x = z → y = z)))
9		ax12olem6.2	. . . 4 ⊢ (¬ ∀x x = y → (y = w → ∀x y = w))
10	3, 4, 8, 9	dvelimhw 1849	. . 3 ⊢ (¬ ∀x x = y → ((¬ ∀x x = z → y = z) → ∀x(¬ ∀x x = z → y = z)))
11	1	19.21h 1797	. . 3 ⊢ (∀x(¬ ∀x x = z → y = z) ↔ (¬ ∀x x = z → ∀x y = z))
12	10, 11	syl6ib 217	. 2 ⊢ (¬ ∀x x = y → ((¬ ∀x x = z → y = z) → (¬ ∀x x = z → ∀x y = z)))
13	12	pm2.86d 93	1 ⊢ (¬ ∀x x = y → (¬ ∀x x = z → (y = z → ∀x y = z)))

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-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

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

This theorem is referenced by:  ax12olem7  1933