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Theorem nfss 3060
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 3059 . 2  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
4 nfra1 2443 . 2  |-  F/ x A. x  e.  A  x  e.  B
53, 4nfxfr 1435 1  |-  F/ x  A  C_  B
Colors of variables: wff set class
Syntax hints:   F/wnf 1421    e. wcel 1465   F/_wnfc 2245   A.wral 2393    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-in 3047  df-ss 3054
This theorem is referenced by:  ssrexf  3129  nfpw  3493  ssiun2s  3827  triun  4009  ssopab2b  4168  nffrfor  4240  tfis  4467  nfrel  4594  nffun  5116  nff  5239  fvmptssdm  5473  ssoprab2b  5796  nfsum1  11093  nfsum  11094
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