Theorem nfss 3194

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 3193	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2539	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1498	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1484   e. wcel 2178   F/_wnfc 2337   A.wral 2486   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-in 3180  df-ss 3187

This theorem is referenced by:  ssrexf  3263  nfpw  3639  ssiun2s  3985  triun  4171  ssopab2b  4341  nffrfor  4413  tfis  4649  nfrel  4778  nffun  5313  nff  5442  fvmptssdm  5687  ssoprab2b  6025  nfsum1  11782  nfsum  11783  nfcprod1  11980  nfcprod  11981