Theorem ax12 1935

Description: Derive ax-12 1925 from ax12v 1926 via ax12o 1934. This shows that the weakening in ax12v 1926 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.)

Assertion

Ref	Expression
ax12	⊢ (¬ x = y → (y = z → ∀x y = z))

Proof of Theorem ax12

Step	Hyp	Ref	Expression
1		sp 1747	. . . . . 6 ⊢ (∀x x = y → x = y)
2	1	con3i 127	. . . . 5 ⊢ (¬ x = y → ¬ ∀x x = y)
3	2	adantr 451	. . . 4 ⊢ ((¬ x = y ∧ y = z) → ¬ ∀x x = y)
4		equtrr 1683	. . . . . . . 8 ⊢ (z = y → (x = z → x = y))
5	4	equcoms 1681	. . . . . . 7 ⊢ (y = z → (x = z → x = y))
6	5	con3rr3 128	. . . . . 6 ⊢ (¬ x = y → (y = z → ¬ x = z))
7	6	imp 418	. . . . 5 ⊢ ((¬ x = y ∧ y = z) → ¬ x = z)
8		sp 1747	. . . . 5 ⊢ (∀x x = z → x = z)
9	7, 8	nsyl 113	. . . 4 ⊢ ((¬ x = y ∧ y = z) → ¬ ∀x x = z)
10		ax12o 1934	. . . 4 ⊢ (¬ ∀x x = y → (¬ ∀x x = z → (y = z → ∀x y = z)))
11	3, 9, 10	sylc 56	. . 3 ⊢ ((¬ x = y ∧ y = z) → (y = z → ∀x y = z))
12	11	ex 423	. 2 ⊢ (¬ x = y → (y = z → (y = z → ∀x y = z)))
13	12	pm2.43d 44	1 ⊢ (¬ x = y → (y = z → ∀x y = z))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀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  ax-12 1925

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

This theorem is referenced by: (None)