Theorem axc11-o 37626

Description: Show that ax-c11 37562 can be derived from ax-c11n 37563 and ax-12 2171. An open problem is whether this theorem can be derived from ax-c11n 37563 and the others when ax-12 2171 is replaced with ax-c15 37564 or ax12v 2172. See Theorems axc11nfromc11 37601 for the rederivation of ax-c11n 37563 from axc11 2428.

Normally, axc11 2428 should be used rather than ax-c11 37562 or axc11-o 37626, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc11-o	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11-o

Step	Hyp	Ref	Expression
1		ax-c11n 37563	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2		ax12 2421	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
3	2	equcoms 2023	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
4	3	sps-o 37583	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑)))
5		pm2.27 42	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑))
6	5	al2imi 1817	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑))
7	1, 4, 6	sylsyld 61	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2370  ax-c5 37558  ax-c11n 37563

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)