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Theorem nfss 3018
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
StepHypRef Expression
1 dfss2f.1 . . 3  |-  F/_ x A
2 dfss2f.2 . . 3  |-  F/_ x B
31, 2dfss3f 3017 . 2  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
4 nfra1 2409 . 2  |-  F/ x A. x  e.  A  x  e.  B
53, 4nfxfr 1408 1  |-  F/ x  A  C_  B
Colors of variables: wff set class
Syntax hints:   F/wnf 1394    e. wcel 1438   F/_wnfc 2215   A.wral 2359    C_ wss 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-in 3005  df-ss 3012
This theorem is referenced by:  nfpw  3442  ssiun2s  3774  triun  3949  ssopab2b  4103  nffrfor  4175  tfis  4398  nfrel  4523  nffun  5038  nff  5158  fvmptssdm  5387  ssoprab2b  5706  nfsum1  10745  nfsum  10746
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