Theorem aevlemOLD 1989

Description: Old proof of aevlem 1981. Obsolete as of 29-Mar-2021. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2246, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable groups:   z, w   x, y

Proof of Theorem aevlemOLD

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvaev 1979	. 2 |- ( A. z z = w -> A. v v = w )
2		ax5d 1840	. . 3 |- ( -. A. z z = v -> ( v = w -> A. z v = w ) )
3	2	axc11nlemOLD2 1988	. 2 |- ( A. v v = w -> A. z z = v )
4		cbvaev 1979	. 2 |- ( A. z z = v -> A. x x = v )
5		ax5d 1840	. . 3 |- ( -. A. y y = x -> ( x = v -> A. y x = v ) )
6	5	axc11nlemOLD2 1988	. 2 |- ( A. x x = v -> A. y y = x )
7	1, 3, 4, 6	4syl 19	1 |- ( A. z z = w -> A. y y = x )

Syntax hints:   -. wn 3   -> 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: (None)