Theorem zfinf 8077

Description: Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)

Assertion

Ref	Expression
zfinf

Distinct variable group:   ,,

Proof of Theorem zfinf

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-inf 8076	. 2
2		elequ1 1821	. . . . . 6
3		elequ1 1821	. . . . . . . 8
4	3	anbi1d 704	. . . . . . 7
5	4	exbidv 1714	. . . . . 6
6	2, 5	imbi12d 320	. . . . 5
7	6	cbvalv 2023	. . . 4
8	7	anbi2i 694	. . 3
9	8	exbii 1667	. 2
10	1, 9	mpbi 208	1

Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  E.wex 1612

This theorem is referenced by:  axinf2  8078  axinfndlem1  9004

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-8 1820  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-inf 8076

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617