Theorem aev 1991

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.)

Assertion

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

Distinct variable group:   x,y

Proof of Theorem aev

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbae 1953	. 2 ⊢ (∀x x = y → ∀z∀x x = y)
2		ax10lem5 1942	. . 3 ⊢ (∀x x = y → ∀u u = v)
3		ax-8 1675	. . . 4 ⊢ (u = w → (u = v → w = v))
4	3	spimv 1990	. . 3 ⊢ (∀u u = v → w = v)
5	2, 4	syl 15	. 2 ⊢ (∀x x = y → w = v)
6	1, 5	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

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:  ax16ALT2  2048  a16gALT  2049