Theorem aev-o 39746

Description: A "distinctor elimination" lemma with no disjoint variable conditions on variables in the consequent, proved without using ax-c16 39707. Version of aev 2092 using ax-c11 39702. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
aev-o	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Distinct variable group:   𝑥,𝑦

Proof of Theorem aev-o

Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbae-o 39718	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2		hbae-o 39718	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑡∀𝑥 𝑥 = 𝑦)
3		ax7 2049	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦))
4	3	spimvw 2019	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑡 = 𝑦)
5	2, 4	alrimih 1857	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑡 𝑡 = 𝑦)
6		ax7 2049	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (𝑦 = 𝑡 → 𝑢 = 𝑡))
7		equcomi 2050	. . . . . . . 8 ⊢ (𝑢 = 𝑡 → 𝑡 = 𝑢)
8	6, 7	syl6 36	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (𝑦 = 𝑡 → 𝑡 = 𝑢))
9	8	spimvw 2019	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑡 → 𝑡 = 𝑢)
10	9	aecoms-o 39717	. . . . 5 ⊢ (∀𝑡 𝑡 = 𝑦 → 𝑡 = 𝑢)
11	10	axc4i-o 39713	. . . 4 ⊢ (∀𝑡 𝑡 = 𝑦 → ∀𝑡 𝑡 = 𝑢)
12		hbae-o 39718	. . . . 5 ⊢ (∀𝑡 𝑡 = 𝑢 → ∀𝑣∀𝑡 𝑡 = 𝑢)
13		ax7 2049	. . . . . 6 ⊢ (𝑡 = 𝑣 → (𝑡 = 𝑢 → 𝑣 = 𝑢))
14	13	spimvw 2019	. . . . 5 ⊢ (∀𝑡 𝑡 = 𝑢 → 𝑣 = 𝑢)
15	12, 14	alrimih 1857	. . . 4 ⊢ (∀𝑡 𝑡 = 𝑢 → ∀𝑣 𝑣 = 𝑢)
16		aecom-o 39716	. . . 4 ⊢ (∀𝑣 𝑣 = 𝑢 → ∀𝑢 𝑢 = 𝑣)
17	11, 15, 16	3syl 19	. . 3 ⊢ (∀𝑡 𝑡 = 𝑦 → ∀𝑢 𝑢 = 𝑣)
18		ax7 2049	. . . 4 ⊢ (𝑢 = 𝑤 → (𝑢 = 𝑣 → 𝑤 = 𝑣))
19	18	spimvw 2019	. . 3 ⊢ (∀𝑢 𝑢 = 𝑣 → 𝑤 = 𝑣)
20	5, 17, 19	3syl 19	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑤 = 𝑣)
21	1, 20	alrimih 1857	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

This proof depends on syntax axioms:   → 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-11 2195  ax-c5 39698  ax-c4 39699  ax-c7 39700  ax-c10 39701  ax-c11 39702  ax-c9 39705

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

This theorem is used by:  axc16g-o  39749