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Theorem ax13fromc9 32709
Description: Derive ax-13 2137 from ax-c9 32695 and other older axioms.

This proof uses newer axioms ax-4 1711 and ax-6 1836, 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 32689 and ax-c10 32691. (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 32688 . . . . . 6 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21con3i 144 . . . . 5 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
32adantr 474 . . . 4 ((¬ 𝑥 = 𝑦𝑦 = 𝑧) → ¬ ∀𝑥 𝑥 = 𝑦)
4 equtrr 1898 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
54equcoms 1896 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 = 𝑧𝑥 = 𝑦))
65con3rr3 145 . . . . . 6 𝑥 = 𝑦 → (𝑦 = 𝑧 → ¬ 𝑥 = 𝑧))
76imp 438 . . . . 5 ((¬ 𝑥 = 𝑦𝑦 = 𝑧) → ¬ 𝑥 = 𝑧)
8 ax-c5 32688 . . . . 5 (∀𝑥 𝑥 = 𝑧𝑥 = 𝑧)
97, 8nsyl 127 . . . 4 ((¬ 𝑥 = 𝑦𝑦 = 𝑧) → ¬ ∀𝑥 𝑥 = 𝑧)
10 ax-c9 32695 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
113, 9, 10sylc 62 . . 3 ((¬ 𝑥 = 𝑦𝑦 = 𝑧) → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
1211ex 443 . 2 𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
1312pm2.43d 50 1 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 378  wal 1466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-c5 32688  ax-c9 32695
This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693
This theorem is referenced by: (None)
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