Theorem nfss 2992

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 2991	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2397	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1403	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1389   e. wcel 1433   F/_wnfc 2206   A.wral 2348   C_ wss 2973

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-in 2979  df-ss 2986

This theorem is referenced by:  nfpw  3394  ssiun2s  3722  triun  3888  ssopab2b  4031  nffrfor  4103  tfis  4324  nfrel  4443  nffun  4944  nff  5063  fvmptssdm  5276  ssoprab2b  5582  nfsum1  10193  nfsum  10194