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Theorem ax13 2388
 Description: Derive ax-13 2385 from ax13v 2386 and Tarski's FOL. This shows that the weakening in ax13v 2386 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage (Revised by Wolf Lammen, 2-Jun-2021.)
Assertion
Ref Expression
ax13 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Proof of Theorem ax13
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equvinv 2029 . . . 4 (𝑦 = 𝑧 ↔ ∃𝑤(𝑤 = 𝑦𝑤 = 𝑧))
2 ax13lem1 2387 . . . . . . . . 9 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
32imp 407 . . . . . . . 8 ((¬ 𝑥 = 𝑦𝑤 = 𝑦) → ∀𝑥 𝑤 = 𝑦)
4 ax13lem1 2387 . . . . . . . . 9 𝑥 = 𝑧 → (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧))
54imp 407 . . . . . . . 8 ((¬ 𝑥 = 𝑧𝑤 = 𝑧) → ∀𝑥 𝑤 = 𝑧)
6 ax7v1 2010 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
76imp 407 . . . . . . . . 9 ((𝑤 = 𝑦𝑤 = 𝑧) → 𝑦 = 𝑧)
87alanimi 1810 . . . . . . . 8 ((∀𝑥 𝑤 = 𝑦 ∧ ∀𝑥 𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧)
93, 5, 8syl2an 595 . . . . . . 7 (((¬ 𝑥 = 𝑦𝑤 = 𝑦) ∧ (¬ 𝑥 = 𝑧𝑤 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
109an4s 656 . . . . . 6 (((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) ∧ (𝑤 = 𝑦𝑤 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
1110ex 413 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → ((𝑤 = 𝑦𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧))
1211exlimdv 1927 . . . 4 ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → (∃𝑤(𝑤 = 𝑦𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧))
131, 12syl5bi 243 . . 3 ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
1413ex 413 . 2 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
15 ax13b 2032 . 2 ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
1614, 15mpbir 232 1 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1528  ∃wex 1773 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-13 2385 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774 This theorem is referenced by:  equvini  2474  equviniOLD  2475  sbequiOLD  2532  sbequiALT  2594
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