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Theorem nfss 2966
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 2965 . 2  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
4 nfra1 2372 . 2  |-  F/ x A. x  e.  A  x  e.  B
53, 4nfxfr 1379 1  |-  F/ x  A  C_  B
Colors of variables: wff set class
Syntax hints:   F/wnf 1365    e. wcel 1409   F/_wnfc 2181   A.wral 2323    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-in 2952  df-ss 2959
This theorem is referenced by:  nfpw  3399  ssiun2s  3729  triun  3895  ssopab2b  4041  nffrfor  4113  tfis  4334  nfrel  4453  nffun  4952  nff  5071  fvmptssdm  5283  ssoprab2b  5590  nfsum1  10106  nfsum  10107
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