Theorem aevlem0 1980

Description: Lemma for aevlem 1981. Instance of aev 1983. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 30-Mar-2021.)

Assertion

Ref	Expression
aevlem0	|- ( A. x x = y -> A. z z = x )

Distinct variable group:   x, y, z

Proof of Theorem aevlem0

Step	Hyp	Ref	Expression
1		spaev 1978	. . 3 |- ( A. x x = y -> x = y )
2	1	alrimiv 1855	. 2 |- ( A. x x = y -> A. z x = y )
3		cbvaev 1979	. 2 |- ( A. x x = y -> A. z z = y )
4		equeuclr 1950	. . 3 |- ( x = y -> ( z = y -> z = x ) )
5	4	al2imi 1743	. 2 |- ( A. z x = y -> ( A. z z = y -> A. z z = x ) )
6	2, 3, 5	sylc 65	1 |- ( A. x x = y -> A. z z = x )

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

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

This theorem is referenced by:  aevlem  1981