Theorem bm1.3ii 4576

Description: Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4573. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)

Hypothesis

Ref	Expression
bm1.3ii.1

Assertion

Ref	Expression
bm1.3ii

Distinct variable groups:   ,   ,

Proof of Theorem bm1.3ii

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bm1.3ii.1	. . . . 5
2		elequ2 1823	. . . . . . . 8
3	2	imbi2d 316	. . . . . . 7
4	3	albidv 1713	. . . . . 6
5	4	cbvexv 2024	. . . . 5
6	1, 5	mpbi 208	. . . 4
7		ax-sep 4573	. . . 4
8	6, 7	pm3.2i 455	. . 3
9	8	exan 1973	. 2
10		19.42v 1775	. . . 4
11		bimsc1 938	. . . . . 6
12	11	alanimi 1637	. . . . 5
13	12	eximi 1656	. . . 4
14	10, 13	sylbir 213	. . 3
15	14	exlimiv 1722	. 2
16	9, 15	ax-mp 5	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612

This theorem is referenced by:  axpow3  4633  pwex  4635  zfpair2  4692  axun2  6594  uniex2  6595

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-sep 4573

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