Theorem nfss 3497

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 3496	. 2 |- ( A C_ B <-> A. x e. A x e. B )
4		nfra1 2845	. 2 |- F/ x A. x e. A x e. B
5	3, 4	nfxfr 1625	1 |- F/ x A C_ B

Syntax hints:   F/wnf 1599   e. wcel 1767   F/_wnfc 2615   A.wral 2814   C_ wss 3476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-in 3483  df-ss 3490

This theorem is referenced by:  nfpw  4022  ssiun2s  4369  triun  4553  ssopab2b  4774  nffr  4853  nfrel  5088  nffun  5610  nff  5727  fvmptss  5959  ssoprab2b  6339  tfis  6674  ovmptss  6865  oawordeulem  7204  nnawordex  7287  r1val1  8205  cardaleph  8471  nfsum1  13478  nfsum  13479  iuncon  19735  ovolfiniun  21739  ovoliunlem3  21742  ovoliun  21743  ovoliun2  21744  ovoliunnul  21745  limciun  22125  ssiun2sf  27197  ssrelf  27236  funimass4f  27244  nfcprod1  28895  nfcprod  28896  nfwrecs  29191  totbndbnd  30115  ssrexf  31193  stoweidlem53  31580  stoweidlem57  31584  iunconlem2  33032  bnj1408  33388