Theorem aev 1943

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 1842. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 1999, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.)

Assertion

Ref	Expression
aev

Distinct variable group:   ,

Proof of Theorem aev

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem1 1939	. . 3
2		ax6ev 1749	. . . 4
3		ax-7 1790	. . . . 5
4	3	aleximi 1653	. . . 4
5	2, 4	mpi 17	. . 3
6		ax5e 1706	. . 3
7	1, 5, 6	3syl 20	. 2
8		axc16g 1940	. 2
9	7, 8	mpd 15	1

Syntax hints:  ->wi 4  A.wal 1393  E.wex 1612

This theorem is referenced by:  ax16nf  1944  axc16gALT  2106  2ax6e  2194  wl-naev  29982  wl-ax11-lem2  30026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-12 1854

This theorem depends on definitions:  df-bi 185  df-ex 1613