Theorem nfss 3140

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 3139	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2501	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1467	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1453   e. wcel 2141   F/_wnfc 2299   A.wral 2448   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-in 3127  df-ss 3134

This theorem is referenced by:  ssrexf  3209  nfpw  3579  ssiun2s  3917  triun  4100  ssopab2b  4261  nffrfor  4333  tfis  4567  nfrel  4696  nffun  5221  nff  5344  fvmptssdm  5580  ssoprab2b  5910  nfsum1  11319  nfsum  11320  nfcprod1  11517  nfcprod  11518