Theorem dfss2f 3265

Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)

Hypotheses

Ref	Expression
dfss2f.1	|- F/_xA
dfss2f.2	|- F/_xB

Assertion

Ref	Expression
dfss2f	|- (A C_ B <-> A.x(x e. A -> x e. B))

Proof of Theorem dfss2f

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3263	. 2 |- (A C_ B <-> A.z(z e. A -> z e. B))
2		dfss2f.1	. . . . 5 |- F/_xA
3	2	nfcri 2484	. . . 4 |- F/x z e. A
4		dfss2f.2	. . . . 5 |- F/_xB
5	4	nfcri 2484	. . . 4 |- F/x z e. B
6	3, 5	nfim 1813	. . 3 |- F/x(z e. A -> z e. B)
7		nfv 1619	. . 3 |- F/z(x e. A -> x e. B)
8		eleq1 2413	. . . 4 |- (z = x -> (z e. A <-> x e. A))
9		eleq1 2413	. . . 4 |- (z = x -> (z e. B <-> x e. B))
10	8, 9	imbi12d 311	. . 3 |- (z = x -> ((z e. A -> z e. B) <-> (x e. A -> x e. B)))
11	6, 7, 10	cbval 1984	. 2 |- (A.z(z e. A -> z e. B) <-> A.x(x e. A -> x e. B))
12	1, 11	bitri 240	1 |- (A C_ B <-> A.x(x e. A -> x e. B))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710   F/_wnfc 2477   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  dfss3f  3266  ss2ab  3335