Theorem ax13 2380

Description: Derive ax-13 2377 from ax13v 2378 and Tarski's FOL. This shows that the weakening in ax13v 2378 is still sufficient for a complete system. Preferably, use the weaker ax13w 2136 to avoid the propagation of ax-13 2377. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 2-Jun-2021.) (New usage is discouraged.)

Assertion

Ref	Expression
ax13	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Proof of Theorem ax13

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinv 2028	. . . 4 ⊢ (𝑦 = 𝑧 ↔ ∃𝑤(𝑤 = 𝑦 ∧ 𝑤 = 𝑧))
2		ax13lem1 2379	. . . . . . . . 9 ⊢ (¬ 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
3	2	imp 406	. . . . . . . 8 ⊢ ((¬ 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → ∀𝑥 𝑤 = 𝑦)
4		ax13lem1 2379	. . . . . . . . 9 ⊢ (¬ 𝑥 = 𝑧 → (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧))
5	4	imp 406	. . . . . . . 8 ⊢ ((¬ 𝑥 = 𝑧 ∧ 𝑤 = 𝑧) → ∀𝑥 𝑤 = 𝑧)
6		ax7v1 2009	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 → 𝑦 = 𝑧))
7	6	imp 406	. . . . . . . . 9 ⊢ ((𝑤 = 𝑦 ∧ 𝑤 = 𝑧) → 𝑦 = 𝑧)
8	7	alanimi 1816	. . . . . . . 8 ⊢ ((∀𝑥 𝑤 = 𝑦 ∧ ∀𝑥 𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧)
9	3, 5, 8	syl2an 596	. . . . . . 7 ⊢ (((¬ 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) ∧ (¬ 𝑥 = 𝑧 ∧ 𝑤 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
10	9	an4s 660	. . . . . 6 ⊢ (((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) ∧ (𝑤 = 𝑦 ∧ 𝑤 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
11	10	ex 412	. . . . 5 ⊢ ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → ((𝑤 = 𝑦 ∧ 𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧))
12	11	exlimdv 1933	. . . 4 ⊢ ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → (∃𝑤(𝑤 = 𝑦 ∧ 𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧))
13	1, 12	biimtrid 242	. . 3 ⊢ ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
14	13	ex 412	. 2 ⊢ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
15		ax13b 2031	. 2 ⊢ ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
16	14, 15	mpbir 231	1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  equvini  2460