Theorem dfss3f 3089

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 3088	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		df-ral 2421	. 2 |- ( A. x e. A x e. B <-> A. x ( x e. A -> x e. B ) )
5	3, 4	bitr4i 186	1 |- ( A C_ B <-> A. x e. A x e. B )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   e. wcel 1480   F/_wnfc 2268   A.wral 2416   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-ral 2421  df-in 3077  df-ss 3084

This theorem is referenced by:  nfss  3090