Theorem nfss 3121

Description: If x is not free in A and B, it is not free in A C_ B. (Contributed by NM, 27-Dec-1996.)

Hypotheses

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

Assertion

Ref	Expression
nfss	|- F/ x A C_ B

Proof of Theorem nfss

Step	Hyp	Ref	Expression
1		dfss2f.1	. . 3 |- F/_ x A
2		dfss2f.2	. . 3 |- F/_ x B
3	1, 2	dfss3f 3120	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2488	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1454	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1440   e. wcel 2128   F/_wnfc 2286   A.wral 2435   C_ wss 3102

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-in 3108  df-ss 3115

This theorem is referenced by:  ssrexf  3190  nfpw  3556  ssiun2s  3893  triun  4075  ssopab2b  4236  nffrfor  4308  tfis  4541  nfrel  4670  nffun  5192  nff  5315  fvmptssdm  5551  ssoprab2b  5875  nfsum1  11248  nfsum  11249  nfcprod1  11446  nfcprod  11447