Theorem nfss 3018

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 3017	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2409	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1408	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1394   e. wcel 1438   F/_wnfc 2215   A.wral 2359   C_ wss 2999

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-in 3005  df-ss 3012

This theorem is referenced by:  nfpw  3442  ssiun2s  3774  triun  3949  ssopab2b  4103  nffrfor  4175  tfis  4398  nfrel  4523  nffun  5038  nff  5158  fvmptssdm  5387  ssoprab2b  5706  nfsum1  10745  nfsum  10746