Theorem nfss 3241

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 3240	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2581	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1527	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1513   e. wcel 2209   F/_wnfc 2379   A.wral 2528   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-in 3226  df-ss 3233

This theorem is referenced by:  ssrexf  3310  nfpw  3701  ssiun2s  4051  triun  4237  ssopab2b  4414  nffrfor  4488  tfis  4725  nfrel  4855  nffun  5395  nff  5525  fvmptssdm  5784  ssoprab2b  6135  funimass4f  6349  nfsum1  12100  nfsum  12101  nfcprod1  12299  nfcprod  12300