Theorem dfss3f 3139

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 3138	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4		df-ral 2453	. 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 1346   e. wcel 2141   F/_wnfc 2299   A.wral 2448   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-in 3127  df-ss 3134

This theorem is referenced by:  nfss  3140