Theorem ax12olem2 1928

Description: Lemma for ax12o 1934. Negate the equalities in ax-12 1925, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.)

Hypothesis

Ref	Expression
ax12olem2.1	⊢ (¬ x = y → (y = w → ∀x y = w))

Assertion

Ref	Expression
ax12olem2	⊢ (¬ x = y → (¬ y = z → ∀x ¬ y = z))

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

Proof of Theorem ax12olem2

Step	Hyp	Ref	Expression
1		ax12olem2.1	. . . . . 6 ⊢ (¬ x = y → (y = w → ∀x y = w))
2	1	anim1d 547	. . . . 5 ⊢ (¬ x = y → ((y = w ∧ ¬ z = w) → (∀x y = w ∧ ¬ z = w)))
3		ax-17 1616	. . . . . . 7 ⊢ (¬ z = w → ∀x ¬ z = w)
4	3	anim2i 552	. . . . . 6 ⊢ ((∀x y = w ∧ ¬ z = w) → (∀x y = w ∧ ∀x ¬ z = w))
5		19.26 1593	. . . . . 6 ⊢ (∀x(y = w ∧ ¬ z = w) ↔ (∀x y = w ∧ ∀x ¬ z = w))
6	4, 5	sylibr 203	. . . . 5 ⊢ ((∀x y = w ∧ ¬ z = w) → ∀x(y = w ∧ ¬ z = w))
7	2, 6	syl6 29	. . . 4 ⊢ (¬ x = y → ((y = w ∧ ¬ z = w) → ∀x(y = w ∧ ¬ z = w)))
8	7	eximdv 1622	. . 3 ⊢ (¬ x = y → (∃w(y = w ∧ ¬ z = w) → ∃w∀x(y = w ∧ ¬ z = w)))
9		19.12 1847	. . 3 ⊢ (∃w∀x(y = w ∧ ¬ z = w) → ∀x∃w(y = w ∧ ¬ z = w))
10	8, 9	syl6 29	. 2 ⊢ (¬ x = y → (∃w(y = w ∧ ¬ z = w) → ∀x∃w(y = w ∧ ¬ z = w)))
11		ax12olem1 1927	. 2 ⊢ (∃w(y = w ∧ ¬ z = w) ↔ ¬ y = z)
12	11	albii 1566	. 2 ⊢ (∀x∃w(y = w ∧ ¬ z = w) ↔ ∀x ¬ y = z)
13	10, 11, 12	3imtr3g 260	1 ⊢ (¬ x = y → (¬ y = z → ∀x ¬ y = z))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀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-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax12olem4  1930