Theorem ax6vsep 4577

Description: Derive a weakened version of ax-6 1747 ( i.e. ax6v 1748), where and must be distinct, from Separation ax-sep 4573 and Extensionality ax-ext 2435. See ax6 2003 for the derivation of ax-6 1747 from ax6v 1748. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6vsep

Distinct variable group:   ,

Proof of Theorem ax6vsep

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 4573	. . 3
2		id 22	. . . . . . . . 9
3	2	biantru 505	. . . . . . . 8
4	3	bibi2i 313	. . . . . . 7
5	4	biimpri 206	. . . . . 6
6	5	alimi 1633	. . . . 5
7		ax-ext 2435	. . . . 5
8	6, 7	syl 16	. . . 4
9	8	eximi 1656	. . 3
10	1, 9	ax-mp 5	. 2
11		df-ex 1613	. 2
12	10, 11	mpbi 208	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818

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-ext 2435  ax-sep 4573

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613