Theorem nfss 3135

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 3134	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2497	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1462	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1448   e. wcel 2136   F/_wnfc 2295   A.wral 2444   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-in 3122  df-ss 3129

This theorem is referenced by:  ssrexf  3204  nfpw  3572  ssiun2s  3910  triun  4093  ssopab2b  4254  nffrfor  4326  tfis  4560  nfrel  4689  nffun  5211  nff  5334  fvmptssdm  5570  ssoprab2b  5899  nfsum1  11297  nfsum  11298  nfcprod1  11495  nfcprod  11496