Theorem dfss2f 3088

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 3086	. 2 |- ( A C_ B <-> A. z ( z e. A -> z e. B ) )
2		dfss2f.1	. . . . 5 |- F/_ x A
3	2	nfcri 2275	. . . 4 |- F/ x z e. A
4		dfss2f.2	. . . . 5 |- F/_ x B
5	4	nfcri 2275	. . . 4 |- F/ x z e. B
6	3, 5	nfim 1551	. . 3 |- F/ x ( z e. A -> z e. B )
7		nfv 1508	. . 3 |- F/ z ( x e. A -> x e. B )
8		eleq1 2202	. . . 4 |- ( z = x -> ( z e. A <-> x e. A ) )
9		eleq1 2202	. . . 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 1727	. 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 1329   e. wcel 1480   F/_wnfc 2268   C_ wss 3071

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084

This theorem is referenced by:  dfss3f  3089  ssrd  3102  ssrmof  3160  ss2ab  3165