Theorem ax13ALT 2362

Description: Alternate proof of ax13 2305 from FOL, sp 2112, and axc9 2313. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax13ALT	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Proof of Theorem ax13ALT

Step	Hyp	Ref	Expression
1		sp 2112	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	con3i 152	. . 3 ⊢ (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3		sp 2112	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → 𝑥 = 𝑧)
4	3	con3i 152	. . 3 ⊢ (¬ 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑧)
5		axc9 2313	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
6	2, 4, 5	syl2im 40	. 2 ⊢ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
7		ax13b 1990	. 2 ⊢ ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
8	6, 7	mpbir 223	1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-10 2080  ax-12 2107  ax-13 2302

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748

This theorem is referenced by: (None)