Theorem nfss 2938

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 2937	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2355	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1363	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1349   e. wcel 1393   F/_wnfc 2165   A.wral 2306   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-in 2924  df-ss 2931

This theorem is referenced by:  nfpw  3371  ssiun2s  3701  triun  3867  ssopab2b  4013  nffrfor  4085  tfis  4306  nfrel  4425  nffun  4924  nff  5043  fvmptssdm  5255  ssoprab2b  5562  nfsum1  9875  nfsum  9876