Theorem nfss 3177

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 3176	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2528	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1488	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1474   e. wcel 2167   F/_wnfc 2326   A.wral 2475   C_ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-in 3163  df-ss 3170

This theorem is referenced by:  ssrexf  3246  nfpw  3619  ssiun2s  3961  triun  4145  ssopab2b  4312  nffrfor  4384  tfis  4620  nfrel  4749  nffun  5282  nff  5407  fvmptssdm  5649  ssoprab2b  5983  nfsum1  11538  nfsum  11539  nfcprod1  11736  nfcprod  11737