Theorem aev 1826

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1828. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem aev

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

Step	Hyp	Ref	Expression
1		hbae 1732	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2		hbae 1732	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑓∀𝑥 𝑥 = 𝑦)
3		ax-8 1518	. . . . 5 ⊢ (𝑥 = 𝑓 → (𝑥 = 𝑦 → 𝑓 = 𝑦))
4	3	spimv 1825	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑓 = 𝑦)
5	2, 4	alrimih 1483	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑓 𝑓 = 𝑦)
6		ax-8 1518	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (𝑦 = 𝑓 → 𝑢 = 𝑓))
7		equcomi 1718	. . . . . . . 8 ⊢ (𝑢 = 𝑓 → 𝑓 = 𝑢)
8	6, 7	syl6 33	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (𝑦 = 𝑓 → 𝑓 = 𝑢))
9	8	spimv 1825	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑓 → 𝑓 = 𝑢)
10	9	alequcoms 1530	. . . . 5 ⊢ (∀𝑓 𝑓 = 𝑦 → 𝑓 = 𝑢)
11	10	a5i 1557	. . . 4 ⊢ (∀𝑓 𝑓 = 𝑦 → ∀𝑓 𝑓 = 𝑢)
12		hbae 1732	. . . . 5 ⊢ (∀𝑓 𝑓 = 𝑢 → ∀𝑣∀𝑓 𝑓 = 𝑢)
13		ax-8 1518	. . . . . 6 ⊢ (𝑓 = 𝑣 → (𝑓 = 𝑢 → 𝑣 = 𝑢))
14	13	spimv 1825	. . . . 5 ⊢ (∀𝑓 𝑓 = 𝑢 → 𝑣 = 𝑢)
15	12, 14	alrimih 1483	. . . 4 ⊢ (∀𝑓 𝑓 = 𝑢 → ∀𝑣 𝑣 = 𝑢)
16		alequcom 1529	. . . 4 ⊢ (∀𝑣 𝑣 = 𝑢 → ∀𝑢 𝑢 = 𝑣)
17	11, 15, 16	3syl 17	. . 3 ⊢ (∀𝑓 𝑓 = 𝑦 → ∀𝑢 𝑢 = 𝑣)
18		ax-8 1518	. . . 4 ⊢ (𝑢 = 𝑤 → (𝑢 = 𝑣 → 𝑤 = 𝑣))
19	18	spimv 1825	. . 3 ⊢ (∀𝑢 𝑢 = 𝑣 → 𝑤 = 𝑣)
20	5, 17, 19	3syl 17	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑤 = 𝑣)
21	1, 20	alrimih 1483	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Syntax hints:   → wi 4  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  ax16  1827  a16g  1878