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Theorem nfss 2966
 Description: If 𝑥 is not free in 𝐴 and 𝐵, it is not free in 𝐴 ⊆ 𝐵. (Contributed by NM, 27-Dec-1996.)
Hypotheses
Ref Expression
dfss2f.1 𝑥𝐴
dfss2f.2 𝑥𝐵
Assertion
Ref Expression
nfss 𝑥 𝐴𝐵

Proof of Theorem nfss
StepHypRef Expression
1 dfss2f.1 . . 3 𝑥𝐴
2 dfss2f.2 . . 3 𝑥𝐵
31, 2dfss3f 2965 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
4 nfra1 2372 . 2 𝑥𝑥𝐴 𝑥𝐵
53, 4nfxfr 1379 1 𝑥 𝐴𝐵
 Colors of variables: wff set class Syntax hints:  Ⅎwnf 1365   ∈ wcel 1409  Ⅎwnfc 2181  ∀wral 2323   ⊆ 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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