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Theorem ax13 2394
Description: Derive ax-13 2391 from ax13v 2392 and Tarski's FOL. This shows that the weakening in ax13v 2392 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 2112 . . . 4 (𝑦 = 𝑧 ↔ ∃𝑤(𝑤 = 𝑦𝑤 = 𝑧))
2 ax13lem1 2393 . . . . . . . . 9 𝑥 = 𝑦 → (𝑤 = 𝑦 → ∀𝑥 𝑤 = 𝑦))
32imp 444 . . . . . . . 8 ((¬ 𝑥 = 𝑦𝑤 = 𝑦) → ∀𝑥 𝑤 = 𝑦)
4 ax13lem1 2393 . . . . . . . . 9 𝑥 = 𝑧 → (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧))
54imp 444 . . . . . . . 8 ((¬ 𝑥 = 𝑧𝑤 = 𝑧) → ∀𝑥 𝑤 = 𝑧)
6 ax7v1 2092 . . . . . . . . . 10 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
76imp 444 . . . . . . . . 9 ((𝑤 = 𝑦𝑤 = 𝑧) → 𝑦 = 𝑧)
87alanimi 1893 . . . . . . . 8 ((∀𝑥 𝑤 = 𝑦 ∧ ∀𝑥 𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧)
93, 5, 8syl2an 495 . . . . . . 7 (((¬ 𝑥 = 𝑦𝑤 = 𝑦) ∧ (¬ 𝑥 = 𝑧𝑤 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
109an4s 904 . . . . . 6 (((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) ∧ (𝑤 = 𝑦𝑤 = 𝑧)) → ∀𝑥 𝑦 = 𝑧)
1110ex 449 . . . . 5 ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → ((𝑤 = 𝑦𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧))
1211exlimdv 2010 . . . 4 ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → (∃𝑤(𝑤 = 𝑦𝑤 = 𝑧) → ∀𝑥 𝑦 = 𝑧))
131, 12syl5bi 232 . . 3 ((¬ 𝑥 = 𝑦 ∧ ¬ 𝑥 = 𝑧) → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
1413ex 449 . 2 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
15 ax13b 2115 . 2 ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
1614, 15mpbir 221 1 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854
This theorem is referenced by:  equvini  2483  sbequi  2512
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