Theorem bj-ax12v3ALT 37292

Description: Alternate proof of bj-ax12v3 37291. Uses axc11r 2400 and axc15 2454 instead of ax-12 2213. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ax12v3ALT	⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable group:   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-ax12v3ALT

Step	Hyp	Ref	Expression
1		ax-5 1940	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
2		axc11r 2400	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
3		ala1 1843	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	1, 2, 3	syl56 37	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	4	a1d 26	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
6		axc15 2454	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
7	5, 6	pm2.61i 184	1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → 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-10 2176  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814

This theorem is referenced by: (None)