Theorem ax12olem4 1930

Description: Lemma for ax12o 1934. Construct an intermediate equivalent to ax-12 1925 from two instances of ax-12 1925. (Contributed by NM, 24-Dec-2015.)

Hypotheses

Ref	Expression
ax12olem4.1	⊢ (¬ x = y → (y = z → ∀x y = z))
ax12olem4.2	⊢ (¬ x = y → (y = w → ∀x y = w))

Assertion

Ref	Expression
ax12olem4	⊢ (¬ x = y → (¬ ∀x ¬ y = z → ∀x y = z))

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

Proof of Theorem ax12olem4

Step	Hyp	Ref	Expression
1		ax12olem4.1	. 2 ⊢ (¬ x = y → (y = z → ∀x y = z))
2		ax12olem4.2	. . 3 ⊢ (¬ x = y → (y = w → ∀x y = w))
3	2	ax12olem2 1928	. 2 ⊢ (¬ x = y → (¬ y = z → ∀x ¬ y = z))
4		ax12olem3 1929	. 2 ⊢ ((¬ x = y → (¬ ∀x ¬ y = z → ∀x y = z)) ↔ ((¬ x = y → (y = z → ∀x y = z)) ∧ (¬ x = y → (¬ y = z → ∀x ¬ y = z))))
5	1, 3, 4	mpbir2an 886	1 ⊢ (¬ x = y → (¬ ∀x ¬ 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-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax12o  1934