Theorem nfss 3095

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 3094	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2469	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1451	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1437   e. wcel 1481   F/_wnfc 2269   A.wral 2417   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-in 3082  df-ss 3089

This theorem is referenced by:  ssrexf  3164  nfpw  3528  ssiun2s  3865  triun  4047  ssopab2b  4206  nffrfor  4278  tfis  4505  nfrel  4632  nffun  5154  nff  5277  fvmptssdm  5513  ssoprab2b  5836  nfsum1  11157  nfsum  11158  nfcprod1  11355  nfcprod  11356