Theorem dfss2f 3133

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 3131	. 2 |- ( A C_ B <-> A. z ( z e. A -> z e. B ) )
2		dfss2f.1	. . . . 5 |- F/_ x A
3	2	nfcri 2302	. . . 4 |- F/ x z e. A
4		dfss2f.2	. . . . 5 |- F/_ x B
5	4	nfcri 2302	. . . 4 |- F/ x z e. B
6	3, 5	nfim 1560	. . 3 |- F/ x ( z e. A -> z e. B )
7		nfv 1516	. . 3 |- F/ z ( x e. A -> x e. B )
8		eleq1 2229	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
9		eleq1 2229	. . . 4 |- ( z = x -> ( z e. B <-> x e. B ) )
10	8, 9	imbi12d 233	. . 3 |- ( z = x -> ( ( z e. A -> z e. B ) <-> ( x e. A -> x e. B ) ) )
11	6, 7, 10	cbval 1742	. 2 |- ( A. z ( z e. A -> z e. B ) <-> A. x ( x e. A -> x e. B ) )
12	1, 11	bitri 183	1 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   e. wcel 2136   F/_wnfc 2295   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129

This theorem is referenced by:  dfss3f  3134  ssrd  3147  ssrmof  3205  ss2ab  3210