Theorem dfss3f 3185

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 3184	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		df-ral 2489	. 2 |- ( A. x e. A x e. B <-> A. x ( x e. A -> x e. B ) )
5	3, 4	bitr4i 187	1 |- ( A C_ B <-> A. x e. A x e. B )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   e. wcel 2176   F/_wnfc 2335   A.wral 2484   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-ral 2489  df-in 3172  df-ss 3179

This theorem is referenced by:  nfss  3186