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Theorem ax13fromc9 33103
Description: Derive ax-13 2137 from ax-c9 33087 and other older axioms.

This proof uses newer axioms ax-4 1713 and ax-6 1838, 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 33081 and ax-c10 33083. (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 33080 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21con3i 148 . . 3 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3 ax-c5 33080 . . . 4 (∀𝑥 𝑥 = 𝑧𝑥 = 𝑧)
43con3i 148 . . 3 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑧)
5 ax-c9 33087 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
62, 4, 5syl2im 39 . 2 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
7 ax13b 1914 . 2 ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
86, 7mpbir 219 1 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-c5 33080  ax-c9 33087
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by: (None)
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