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Theorem ax12o 32708
Description: Rederivation of axiom ax-12 1983 from ax-c15 32694, ax-c11 32692, and other older axioms. See theorem axc15 2187 for the derivation of ax-c15 32694 from ax-12 1983.

An open problem is whether we can prove this using ax-c11n 32693 instead of ax-c11 32692.

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, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax12o (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem ax12o
StepHypRef Expression
1 biidd 247 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜑))
21dral1-o 32707 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
3 ax-1 6 . . . . 5 (𝜑 → (𝑥 = 𝑦𝜑))
43alimi 1713 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
52, 4syl6bir 239 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
65a1d 25 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
7 ax-c5 32688 . . 3 (∀𝑦𝜑𝜑)
8 ax-c15 32694 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
97, 8syl7 71 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
106, 9pm2.61i 171 1 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  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-11 1970  ax-c5 32688  ax-c4 32689  ax-c7 32690  ax-c11 32692  ax-c15 32694  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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