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Theorem ax13fromc9 34510
Description: Derive ax-13 2282 from ax-c9 34494 and other older axioms.

This proof uses newer axioms ax-4 1777 and ax-6 1945, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 34488 and ax-c10 34490. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax13fromc9 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Proof of Theorem ax13fromc9
StepHypRef Expression
1 ax-c5 34487 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21con3i 150 . . 3 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3 ax-c5 34487 . . . 4 (∀𝑥 𝑥 = 𝑧𝑥 = 𝑧)
43con3i 150 . . 3 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑧)
5 ax-c9 34494 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
62, 4, 5syl2im 40 . 2 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
7 ax13b 2006 . 2 ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
86, 7mpbir 221 1 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-c5 34487  ax-c9 34494
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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