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Theorem nfss 3058
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 3057 . 2  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
4 nfra1 2441 . 2  |-  F/ x A. x  e.  A  x  e.  B
53, 4nfxfr 1433 1  |-  F/ x  A  C_  B
Colors of variables: wff set class
Syntax hints:   F/wnf 1419    e. wcel 1463   F/_wnfc 2243   A.wral 2391    C_ wss 3039
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-in 3045  df-ss 3052
This theorem is referenced by:  ssrexf  3127  nfpw  3491  ssiun2s  3825  triun  4007  ssopab2b  4166  nffrfor  4238  tfis  4465  nfrel  4592  nffun  5114  nff  5237  fvmptssdm  5471  ssoprab2b  5794  nfsum1  11065  nfsum  11066
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