Theorem aev-o 2182

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 2144. Version of aev 1991 using ax-10o 2139. (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 2153	. 2 ⊢ (∀x x = y → ∀z∀x x = y)
2		hbae-o 2153	. . . 4 ⊢ (∀x x = y → ∀t∀x x = y)
3		ax-8 1675	. . . . 5 ⊢ (x = t → (x = y → t = y))
4	3	spimv 1990	. . . 4 ⊢ (∀x x = y → t = y)
5	2, 4	alrimih 1565	. . 3 ⊢ (∀x x = y → ∀t t = y)
6		ax-8 1675	. . . . . . . 8 ⊢ (y = u → (y = t → u = t))
7		equcomi 1679	. . . . . . . 8 ⊢ (u = t → t = u)
8	6, 7	syl6 29	. . . . . . 7 ⊢ (y = u → (y = t → t = u))
9	8	spimv 1990	. . . . . 6 ⊢ (∀y y = t → t = u)
10	9	aecoms-o 2152	. . . . 5 ⊢ (∀t t = y → t = u)
11	10	a5i-o 2150	. . . 4 ⊢ (∀t t = y → ∀t t = u)
12		hbae-o 2153	. . . . 5 ⊢ (∀t t = u → ∀v∀t t = u)
13		ax-8 1675	. . . . . 6 ⊢ (t = v → (t = u → v = u))
14	13	spimv 1990	. . . . 5 ⊢ (∀t t = u → v = u)
15	12, 14	alrimih 1565	. . . 4 ⊢ (∀t t = u → ∀v v = u)
16		aecom-o 2151	. . . 4 ⊢ (∀v v = u → ∀u u = v)
17	11, 15, 16	3syl 18	. . 3 ⊢ (∀t t = y → ∀u u = v)
18		ax-8 1675	. . . 4 ⊢ (u = w → (u = v → w = v))
19	18	spimv 1990	. . 3 ⊢ (∀u u = v → w = v)
20	5, 17, 19	3syl 18	. 2 ⊢ (∀x x = y → w = v)
21	1, 20	alrimih 1565	1 ⊢ (∀x x = y → ∀z w = v)

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  a16g-o  2186