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Theorem nfci 2906
Description: Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfci.1 𝑥 𝑦𝐴
Assertion
Ref Expression
nfci 𝑥𝐴
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfci
StepHypRef Expression
1 df-nfc 2905 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfci.1 . 2 𝑥 𝑦𝐴
31, 2mpgbir 1877 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wnf 1859  wcel 2148  wnfc 2903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873
This theorem depends on definitions:  df-bi 198  df-nfc 2905
This theorem is referenced by:  nfcii  2907  nfcv  2916  nfab1  2918  nfab  2921  fpwrelmap  29865  esumfzf  30488  bj-nfab1  33137  fsumiunss  40331  climsuse  40364  climinff  40367  fnlimfvre  40430  limsupre3uzlem  40491  pimdecfgtioc  41451  pimincfltioc  41452  smfmullem4  41527  smflimsupmpt  41561
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