Theorem nfss 3186

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 3185	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2537	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1497	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1483   e. wcel 2176   F/_wnfc 2335   A.wral 2484   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-in 3172  df-ss 3179

This theorem is referenced by:  ssrexf  3255  nfpw  3629  ssiun2s  3971  triun  4156  ssopab2b  4324  nffrfor  4396  tfis  4632  nfrel  4761  nffun  5295  nff  5424  fvmptssdm  5666  ssoprab2b  6004  nfsum1  11700  nfsum  11701  nfcprod1  11898  nfcprod  11899