Theorem nfss 3148

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 3147	. 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 3129

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 3135  df-ss 3142

This theorem is referenced by:  ssrexf  3217  nfpw  3588  ssiun2s  3930  triun  4113  ssopab2b  4275  nffrfor  4347  tfis  4581  nfrel  4710  nffun  5238  nff  5361  fvmptssdm  5599  ssoprab2b  5929  nfsum1  11357  nfsum  11358  nfcprod1  11555  nfcprod  11556