Theorem dfss3f 2991

Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)

Hypotheses

Ref	Expression
dfss2f.1	|- F/_ x A
dfss2f.2	|- F/_ x B

Assertion

Ref	Expression
dfss3f	|- ( A C_ B <-> A. x e. A x e. B )

Proof of Theorem dfss3f

Step	Hyp	Ref	Expression
1		dfss2f.1	. . 3 |- F/_ x A
2		dfss2f.2	. . 3 |- F/_ x B
3	1, 2	dfss2f 2990	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		df-ral 2353	. 2 |- ( A. x e. A x e. B <-> A. x ( x e. A -> x e. B ) )
5	3, 4	bitr4i 185	1 |- ( A C_ B <-> A. x e. A x e. B )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   e. wcel 1433   F/_wnfc 2206   A.wral 2348   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-in 2979  df-ss 2986

This theorem is referenced by:  nfss  2992