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Theorem nfss 3813
Description: If 𝑥 is not free in 𝐴 and 𝐵, it is not free in 𝐴𝐵. (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 3812 . 2 (𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
4 nfra1 3122 . 2 𝑥𝑥𝐴 𝑥𝐵
53, 4nfxfr 1897 1 𝑥 𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wnf 1827  wcel 2106  wnfc 2918  wral 3089  wss 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-in 3798  df-ss 3805
This theorem is referenced by:  ssrexf  3883  nfpw  4392  ssiun2s  4797  triun  5000  iunopeqop  5218  ssopab2b  5239  nffr  5329  nfrel  5452  nffun  6158  nff  6287  fvmptss  6553  ssoprab2b  6989  tfis  7332  ovmptss  7539  nfwrecs  7691  oawordeulem  7918  nnawordex  8001  r1val1  8946  cardaleph  9245  nfsum1  14828  nfsum  14829  nfcprod1  15043  nfcprod  15044  iunconn  21640  ovolfiniun  23705  ovoliunlem3  23708  ovoliun  23709  ovoliun2  23710  ovoliunnul  23711  limciun  24095  ssiun2sf  29940  ssrelf  29990  funimass4f  30002  fsumiunle  30139  prodindf  30683  esumiun  30754  bnj1408  31703  nffrecs  32367  totbndbnd  34196  ss2iundf  38890  iunconnlem2  40086  iinssdf  40236  rnmptssbi  40370  stoweidlem53  41179  stoweidlem57  41183  meaiunincf  41606  meaiuninc3  41608  opnvonmbllem2  41756  smflim  41894  nfsetrecs  43520  setrec2fun  43526
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