Theorem ax12b 1689

Description: Two equivalent ways of expressing ax-12 1925. See the comment for ax-12 1925. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2017.)

Assertion

Ref	Expression
ax12b	⊢ ((¬ x = y → (y = z → ∀x y = z)) ↔ (¬ x = y → (¬ x = z → (y = z → ∀x y = z))))

Proof of Theorem ax12b

Step	Hyp	Ref	Expression
1		id 19	. . 3 ⊢ ((¬ x = y → (y = z → ∀x y = z)) → (¬ x = y → (y = z → ∀x y = z)))
2	1	a1dd 42	. 2 ⊢ ((¬ x = y → (y = z → ∀x y = z)) → (¬ x = y → (¬ x = z → (y = z → ∀x y = z))))
3		equtrr 1683	. . . . . 6 ⊢ (z = y → (x = z → x = y))
4	3	equcoms 1681	. . . . 5 ⊢ (y = z → (x = z → x = y))
5	4	con3rr3 128	. . . 4 ⊢ (¬ x = y → (y = z → ¬ x = z))
6		id 19	. . . . . 6 ⊢ ((¬ x = y → (¬ x = z → (y = z → ∀x y = z))) → (¬ x = y → (¬ x = z → (y = z → ∀x y = z))))
7	6	com4l 78	. . . . 5 ⊢ (¬ x = y → (¬ x = z → (y = z → ((¬ x = y → (¬ x = z → (y = z → ∀x y = z))) → ∀x y = z))))
8	7	com23 72	. . . 4 ⊢ (¬ x = y → (y = z → (¬ x = z → ((¬ x = y → (¬ x = z → (y = z → ∀x y = z))) → ∀x y = z))))
9	5, 8	mpdd 36	. . 3 ⊢ (¬ x = y → (y = z → ((¬ x = y → (¬ x = z → (y = z → ∀x y = z))) → ∀x y = z)))
10	9	com3r 73	. 2 ⊢ ((¬ x = y → (¬ x = z → (y = z → ∀x y = z))) → (¬ x = y → (y = z → ∀x y = z)))
11	2, 10	impbii 180	1 ⊢ ((¬ x = y → (y = z → ∀x y = z)) ↔ (¬ x = y → (¬ x = z → (y = z → ∀x y = z))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀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

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

This theorem is referenced by: (None)