Theorem aev 1693

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1695. (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 1606	. 2 |- ( A. x x = y -> A. z A. x x = y )
2		hbae 1606	. . . 4 |- ( A. x x = y -> A. f A. x x = y )
3		ax-8 1395	. . . . 5 |- ( x = f -> ( x = y -> f = y ) )
4	3	spimv 1692	. . . 4 |- ( A. x x = y -> f = y )
5	2, 4	alrimih 1358	. . 3 |- ( A. x x = y -> A. f f = y )
6		ax-8 1395	. . . . . . . 8 |- ( y = u -> ( y = f -> u = f ) )
7		equcomi 1592	. . . . . . . 8 |- ( u = f -> f = u )
8	6, 7	syl6 29	. . . . . . 7 |- ( y = u -> ( y = f -> f = u ) )
9	8	spimv 1692	. . . . . 6 |- ( A. y y = f -> f = u )
10	9	alequcoms 1409	. . . . 5 |- ( A. f f = y -> f = u )
11	10	a5i 1435	. . . 4 |- ( A. f f = y -> A. f f = u )
12		hbae 1606	. . . . 5 |- ( A. f f = u -> A. v A. f f = u )
13		ax-8 1395	. . . . . 6 |- ( f = v -> ( f = u -> v = u ) )
14	13	spimv 1692	. . . . 5 |- ( A. f f = u -> v = u )
15	12, 14	alrimih 1358	. . . 4 |- ( A. f f = u -> A. v v = u )
16		alequcom 1408	. . . 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 1395	. . . 4 |- ( u = w -> ( u = v -> w = v ) )
19	18	spimv 1692	. . 3 |- ( A. u u = v -> w = v )
20	5, 17, 19	3syl 17	. 2 |- ( A. x x = y -> w = v )
21	1, 20	alrimih 1358	1 |- ( A. x x = y -> A. z w = v )

Syntax hints:   -> wi 4   A.wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  ax16  1694  a16g  1744