Theorem ax12 2420

Description: Rederivation of Axiom ax-12 2169 from ax12v 2170 (used only via sp 2174), axc11r 2363, and axc15 2419 (on top of Tarski's FOL). Since this version depends on ax-13 2369, usage of the weaker ax12v 2170, ax12w 2127, ax12i 1968 are preferred. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.) (New usage is discouraged.)

Assertion

Ref	Expression
ax12	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Proof of Theorem ax12

Step	Hyp	Ref	Expression
1		axc11r 2363	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
2		ala1 1813	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	syl6 35	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	a1d 25	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
5		sp 2174	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
6		axc15 2419	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
7	5, 6	syl7 74	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
8	4, 7	pm2.61i 182	1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169  ax-13 2369

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-nf 1784

This theorem is referenced by:  equs5a  2454  equs5e  2455  bj-ax12v3  35868  wl-axc11r  36704  axc11-o  38126