Theorem aev-o 33234

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-c16 33195. Version of aev 1970 using ax-c11 33190. (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 33206	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2		hbae-o 33206	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑡∀𝑥 𝑥 = 𝑦)
3		ax7 1930	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦))
4	3	spimv 2245	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑡 = 𝑦)
5	2, 4	alrimih 1741	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑡 𝑡 = 𝑦)
6		ax7 1930	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (𝑦 = 𝑡 → 𝑢 = 𝑡))
7		equcomi 1931	. . . . . . . 8 ⊢ (𝑢 = 𝑡 → 𝑡 = 𝑢)
8	6, 7	syl6 34	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (𝑦 = 𝑡 → 𝑡 = 𝑢))
9	8	spimv 2245	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑡 → 𝑡 = 𝑢)
10	9	aecoms-o 33205	. . . . 5 ⊢ (∀𝑡 𝑡 = 𝑦 → 𝑡 = 𝑢)
11	10	axc4i-o 33201	. . . 4 ⊢ (∀𝑡 𝑡 = 𝑦 → ∀𝑡 𝑡 = 𝑢)
12		hbae-o 33206	. . . . 5 ⊢ (∀𝑡 𝑡 = 𝑢 → ∀𝑣∀𝑡 𝑡 = 𝑢)
13		ax7 1930	. . . . . 6 ⊢ (𝑡 = 𝑣 → (𝑡 = 𝑢 → 𝑣 = 𝑢))
14	13	spimv 2245	. . . . 5 ⊢ (∀𝑡 𝑡 = 𝑢 → 𝑣 = 𝑢)
15	12, 14	alrimih 1741	. . . 4 ⊢ (∀𝑡 𝑡 = 𝑢 → ∀𝑣 𝑣 = 𝑢)
16		aecom-o 33204	. . . 4 ⊢ (∀𝑣 𝑣 = 𝑢 → ∀𝑢 𝑢 = 𝑣)
17	11, 15, 16	3syl 18	. . 3 ⊢ (∀𝑡 𝑡 = 𝑦 → ∀𝑢 𝑢 = 𝑣)
18		ax7 1930	. . . 4 ⊢ (𝑢 = 𝑤 → (𝑢 = 𝑣 → 𝑤 = 𝑣))
19	18	spimv 2245	. . 3 ⊢ (∀𝑢 𝑢 = 𝑣 → 𝑤 = 𝑣)
20	5, 17, 19	3syl 18	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑤 = 𝑣)
21	1, 20	alrimih 1741	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-11 2021  ax-12 2034  ax-13 2234  ax-c5 33186  ax-c4 33187  ax-c7 33188  ax-c10 33189  ax-c11 33190  ax-c9 33193

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  axc16g-o  33237