Theorem axpweq 4629

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4630 is not used by the proof. (Contributed by NM, 22-Jun-2009.)

Hypothesis

Ref	Expression
axpweq.1

Assertion

Ref	Expression
axpweq

Distinct variable group:   ,,,

Proof of Theorem axpweq

Step	Hyp	Ref	Expression
1		pwidg 4025	. . . 4
2		pweq 4015	. . . . . 6
3	2	eleq2d 2527	. . . . 5
4	3	spcegv 3195	. . . 4
5	1, 4	mpd 15	. . 3
6		elex 3118	. . . 4
7	6	exlimiv 1722	. . 3
8	5, 7	impbii 188	. 2
9		vex 3112	. . . . 5
10	9	elpw2 4616	. . . 4
11		pwss 4027	. . . . 5
12		dfss2 3492	. . . . . . 7
13	12	imbi1i 325	. . . . . 6
14	13	albii 1640	. . . . 5
15	11, 14	bitri 249	. . . 4
16	10, 15	bitri 249	. . 3
17	16	exbii 1667	. 2
18	8, 17	bitri 249	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  cvv 3109  C_wss 3475  ~Pcpw 4012

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3482  df-ss 3489  df-pw 4014