Theorem aev-o 34216

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-c16 34177. Version of aev 1983 using ax-c11 34172. (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	|- ( A. x x = y -> A. 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 34188	. 2 |- ( A. x x = y -> A. z A. x x = y )
2		hbae-o 34188	. . . 4 |- ( A. x x = y -> A. t A. x x = y )
3		ax7 1943	. . . . 5 |- ( x = t -> ( x = y -> t = y ) )
4	3	spimv 2257	. . . 4 |- ( A. x x = y -> t = y )
5	2, 4	alrimih 1751	. . 3 |- ( A. x x = y -> A. t t = y )
6		ax7 1943	. . . . . . . 8 |- ( y = u -> ( y = t -> u = t ) )
7		equcomi 1944	. . . . . . . 8 |- ( u = t -> t = u )
8	6, 7	syl6 35	. . . . . . 7 |- ( y = u -> ( y = t -> t = u ) )
9	8	spimv 2257	. . . . . 6 |- ( A. y y = t -> t = u )
10	9	aecoms-o 34187	. . . . 5 |- ( A. t t = y -> t = u )
11	10	axc4i-o 34183	. . . 4 |- ( A. t t = y -> A. t t = u )
12		hbae-o 34188	. . . . 5 |- ( A. t t = u -> A. v A. t t = u )
13		ax7 1943	. . . . . 6 |- ( t = v -> ( t = u -> v = u ) )
14	13	spimv 2257	. . . . 5 |- ( A. t t = u -> v = u )
15	12, 14	alrimih 1751	. . . 4 |- ( A. t t = u -> A. v v = u )
16		aecom-o 34186	. . . 4 |- ( A. v v = u -> A. u u = v )
17	11, 15, 16	3syl 18	. . 3 |- ( A. t t = y -> A. u u = v )
18		ax7 1943	. . . 4 |- ( u = w -> ( u = v -> w = v ) )
19	18	spimv 2257	. . 3 |- ( A. u u = v -> w = v )
20	5, 17, 19	3syl 18	. 2 |- ( A. x x = y -> w = v )
21	1, 20	alrimih 1751	1 |- ( A. x x = y -> A. z w = v )

Syntax hints:   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-11 2034  ax-12 2047  ax-13 2246  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  axc16g-o  34219