Theorem ax12olem7 1933

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

Hypotheses

Ref	Expression
ax12olem7.1	⊢ (¬ x = z → (¬ ∀x ¬ z = w → ∀x z = w))
ax12olem7.2	⊢ (¬ x = y → (¬ ∀x ¬ y = w → ∀x y = w))

Assertion

Ref	Expression
ax12olem7	⊢ (¬ ∀x x = y → (¬ ∀x x = z → (y = z → ∀x y = z)))

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

Proof of Theorem ax12olem7

Step	Hyp	Ref	Expression
1		ax12olem7.1	. . 3 ⊢ (¬ x = z → (¬ ∀x ¬ z = w → ∀x z = w))
2	1	ax12olem5 1931	. 2 ⊢ (¬ ∀x x = z → (z = w → ∀x z = w))
3		ax12olem7.2	. . 3 ⊢ (¬ x = y → (¬ ∀x ¬ y = w → ∀x y = w))
4	3	ax12olem5 1931	. 2 ⊢ (¬ ∀x x = y → (y = w → ∀x y = w))
5	2, 4	ax12olem6 1932	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:  ax12o  1934