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Theorem ax12 2291
 Description: Rederivation of axiom ax-12 2033 from ax12v 2034, axc11r 2174, and other axioms. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.)
Assertion
Ref Expression
ax12 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem ax12
StepHypRef Expression
1 axc11r 2174 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
2 ala1 1754 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
31, 2syl6 34 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
43a1d 25 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
5 sp 2040 . . 3 (∀𝑦𝜑𝜑)
6 axc15 2290 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
75, 6syl7 71 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
84, 7pm2.61i 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-10 2005  ax-12 2033  ax-13 2233 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700 This theorem is referenced by:  equs5a  2335  equs5e  2336  bj-ax12v3  31655  axc11-o  33037
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