Theorem dfss2f 3594

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/_ x A
dfss2f.2	|- F/_ x B

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 3591	. 2 |- ( A C_ B <-> A. z ( z e. A -> z e. B ) )
2		dfss2f.1	. . . . 5 |- F/_ x A
3	2	nfcri 2758	. . . 4 |- F/ x z e. A
4		dfss2f.2	. . . . 5 |- F/_ x B
5	4	nfcri 2758	. . . 4 |- F/ x z e. B
6	3, 5	nfim 1825	. . 3 |- F/ x ( z e. A -> z e. B )
7		nfv 1843	. . 3 |- F/ z ( x e. A -> x e. B )
8		eleq1 2689	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
9		eleq1 2689	. . . 4 |- ( z = x -> ( z e. B <-> x e. B ) )
10	8, 9	imbi12d 334	. . 3 |- ( z = x -> ( ( z e. A -> z e. B ) <-> ( x e. A -> x e. B ) ) )
11	6, 7, 10	cbval 2271	. 2 |- ( A. z ( z e. A -> z e. B ) <-> A. x ( x e. A -> x e. B ) )
12	1, 11	bitri 264	1 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   F/_wnfc 2751   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588

This theorem is referenced by:  dfss3f  3595  ssrd  3608  ss2ab  3670  rankval4  8730  ssrmo  29334  rabexgfGS  29340  ballotth  30599  dvcosre  40126  itgsinexplem1  40169