Theorem aev 1800

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1802. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

Ref	Expression
aev	|- ( A. x x = y -> A. 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 1706	. 2 |- ( A. x x = y -> A. z A. x x = y )
2		hbae 1706	. . . 4 |- ( A. x x = y -> A. f A. x x = y )
3		ax-8 1492	. . . . 5 |- ( x = f -> ( x = y -> f = y ) )
4	3	spimv 1799	. . . 4 |- ( A. x x = y -> f = y )
5	2, 4	alrimih 1457	. . 3 |- ( A. x x = y -> A. f f = y )
6		ax-8 1492	. . . . . . . 8 |- ( y = u -> ( y = f -> u = f ) )
7		equcomi 1692	. . . . . . . 8 |- ( u = f -> f = u )
8	6, 7	syl6 33	. . . . . . 7 |- ( y = u -> ( y = f -> f = u ) )
9	8	spimv 1799	. . . . . 6 |- ( A. y y = f -> f = u )
10	9	alequcoms 1504	. . . . 5 |- ( A. f f = y -> f = u )
11	10	a5i 1531	. . . 4 |- ( A. f f = y -> A. f f = u )
12		hbae 1706	. . . . 5 |- ( A. f f = u -> A. v A. f f = u )
13		ax-8 1492	. . . . . 6 |- ( f = v -> ( f = u -> v = u ) )
14	13	spimv 1799	. . . . 5 |- ( A. f f = u -> v = u )
15	12, 14	alrimih 1457	. . . 4 |- ( A. f f = u -> A. v v = u )
16		alequcom 1503	. . . 4 |- ( A. v v = u -> A. u u = v )
17	11, 15, 16	3syl 17	. . 3 |- ( A. f f = y -> A. u u = v )
18		ax-8 1492	. . . 4 |- ( u = w -> ( u = v -> w = v ) )
19	18	spimv 1799	. . 3 |- ( A. u u = v -> w = v )
20	5, 17, 19	3syl 17	. 2 |- ( A. x x = y -> w = v )
21	1, 20	alrimih 1457	1 |- ( A. x x = y -> A. z w = v )

Syntax hints:   -> wi 4   A.wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  ax16  1801  a16g  1852