Theorem ax12fromc15 39657

Description: Rederivation of Axiom ax-12 2213 from ax-c15 39641, ax-c11 39639 (used through dral1-o 39656), and other older axioms. See Theorem axc15 2454 for the derivation of ax-c15 39641 from ax-12 2213.

An open problem is whether we can prove this using ax-c11n 39640 instead of ax-c11 39639.

This proof uses newer axioms ax-4 1839 and ax-6 1997, 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 39636 and ax-c10 39638. (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 265	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
2	1	dral1-o 39656	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
3		ax-1 6	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑))
4	3	alimi 1841	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	2, 4	biimtrrdi 257	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	5	a1d 26	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
7		ax-c5 39635	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
8		ax-c15 39641	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
9	7, 8	syl7 75	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
10	6, 9	pm2.61i 184	1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-11 2192  ax-c5 39635  ax-c4 39636  ax-c7 39637  ax-c10 39638  ax-c11 39639  ax-c15 39641  ax-c9 39642

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by: (None)