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Theorem ax12fromc15 33004
Description: Rederivation of axiom ax-12 2033 from ax-c15 32988, ax-c11 32986 (used through dral1-o 33003), and other older axioms. See theorem axc15 2290 for the derivation of ax-c15 32988 from ax-12 2033.

An open problem is whether we can prove this using ax-c11n 32987 instead of ax-c11 32986.

This proof uses newer axioms ax-4 1727 and ax-6 1874, 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 32983 and ax-c10 32985. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax12fromc15 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem ax12fromc15
StepHypRef Expression
1 biidd 250 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜑))
21dral1-o 33003 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
3 ax-1 6 . . . . 5 (𝜑 → (𝑥 = 𝑦𝜑))
43alimi 1729 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
52, 4syl6bir 242 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
65a1d 25 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
7 ax-c5 32982 . . 3 (∀𝑦𝜑𝜑)
8 ax-c15 32988 . . 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 174 1 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
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 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-11 2020  ax-c5 32982  ax-c4 32983  ax-c7 32984  ax-c10 32985  ax-c11 32986  ax-c15 32988  ax-c9 32989
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by: (None)
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