Theorem ax13ALT 2305

Description: Alternate proof of ax13 2249 from FOL, sp 2053, and axc9 2302. (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 2053	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	con3i 150	. . 3 ⊢ (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3		sp 2053	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → 𝑥 = 𝑧)
4	3	con3i 150	. . 3 ⊢ (¬ 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑧)
5		axc9 2302	. . 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 1964	. 2 ⊢ ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
8	6, 7	mpbir 221	1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)