Theorem dfss2f 3192

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		ssalel 3189	. 2 |- ( A C_ B <-> A. z ( z e. A -> z e. B ) )
2		dfss2f.1	. . . . 5 |- F/_ x A
3	2	nfcri 2344	. . . 4 |- F/ x z e. A
4		dfss2f.2	. . . . 5 |- F/_ x B
5	4	nfcri 2344	. . . 4 |- F/ x z e. B
6	3, 5	nfim 1596	. . 3 |- F/ x ( z e. A -> z e. B )
7		nfv 1552	. . 3 |- F/ z ( x e. A -> x e. B )
8		eleq1 2270	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
9		eleq1 2270	. . . 4 |- ( z = x -> ( z e. B <-> x e. B ) )
10	8, 9	imbi12d 234	. . 3 |- ( z = x -> ( ( z e. A -> z e. B ) <-> ( x e. A -> x e. B ) ) )
11	6, 7, 10	cbval 1778	. 2 |- ( A. z ( z e. A -> z e. B ) <-> A. x ( x e. A -> x e. B ) )
12	1, 11	bitri 184	1 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   e. wcel 2178   F/_wnfc 2337   C_ wss 3174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187

This theorem is referenced by:  dfss3f  3193  ssrd  3206  ssrmof  3264  ss2ab  3269