Theorem ax12 2458

Description: Rederivation of Axiom ax-12 2216 from ax12v 2217 (used only via sp 2222), axc11r 2403, and axc15 2457 (on top of Tarski's FOL). Since this version depends on ax-13 2407, usage of the weaker ax12v 2217, ax12w 2171, ax12i 1999 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 2403	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
2		ala1 1846	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	syl6 36	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	a1d 26	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
5		sp 2222	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
6		axc15 2457	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
7	5, 6	syl7 75	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
8	4, 7	pm2.61i 184	1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  equs5a  2492  equs5e  2493  bj-ax12v3  37351  wl-axc11r  38226  axc11-o  39766