Theorem dfss2f 3184

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 3181	. 2 |- ( A C_ B <-> A. z ( z e. A -> z e. B ) )
2		dfss2f.1	. . . . 5 |- F/_ x A
3	2	nfcri 2342	. . . 4 |- F/ x z e. A
4		dfss2f.2	. . . . 5 |- F/_ x B
5	4	nfcri 2342	. . . 4 |- F/ x z e. B
6	3, 5	nfim 1595	. . 3 |- F/ x ( z e. A -> z e. B )
7		nfv 1551	. . 3 |- F/ z ( x e. A -> x e. B )
8		eleq1 2268	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
9		eleq1 2268	. . . 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 1777	. 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 2176   F/_wnfc 2335   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179

This theorem is referenced by:  dfss3f  3185  ssrd  3198  ssrmof  3256  ss2ab  3261