Theorem nfss 3150

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 3149	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2508	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1474	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1460   e. wcel 2148   F/_wnfc 2306   A.wral 2455   C_ wss 3131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-in 3137  df-ss 3144

This theorem is referenced by:  ssrexf  3219  nfpw  3590  ssiun2s  3932  triun  4116  ssopab2b  4278  nffrfor  4350  tfis  4584  nfrel  4713  nffun  5241  nff  5364  fvmptssdm  5603  ssoprab2b  5935  nfsum1  11367  nfsum  11368  nfcprod1  11565  nfcprod  11566