Theorem ax12fromc15 39404

Description: Rederivation of Axiom ax-12 2189 from ax-c15 39388, ax-c11 39386 (used through dral1-o 39403), and other older axioms. See Theorem axc15 2430 for the derivation of ax-c15 39388 from ax-12 2189.

An open problem is whether we can prove this using ax-c11n 39387 instead of ax-c11 39386.

This proof uses newer axioms ax-4 1816 and ax-6 1974, 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 39383 and ax-c10 39385. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax12fromc15	⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Proof of Theorem ax12fromc15

Step	Hyp	Ref	Expression
1		biidd 263	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
2	1	dral1-o 39403	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
3		ax-1 6	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑))
4	3	alimi 1818	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	2, 4	biimtrrdi 255	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	5	a1d 25	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
7		ax-c5 39382	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
8		ax-c15 39388	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
9	7, 8	syl7 74	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
10	6, 9	pm2.61i 183	1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-11 2168  ax-c5 39382  ax-c4 39383  ax-c7 39384  ax-c10 39385  ax-c11 39386  ax-c15 39388  ax-c9 39389

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by: (None)