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Theorem nfci 2938
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 2937 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfci.1 . 2 𝑥 𝑦𝐴
31, 2mpgbir 1881 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wnf 1863  wcel 2156  wnfc 2935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877
This theorem depends on definitions:  df-bi 198  df-nfc 2937
This theorem is referenced by:  nfcii  2939  nfcv  2948  nfab1  2950  nfab  2953  fpwrelmap  29835  esumfzf  30456  bj-nfab1  33099  fsumiunss  40287  climsuse  40320  climinff  40323  fnlimfvre  40386  limsupre3uzlem  40447  pimdecfgtioc  41407  pimincfltioc  41408  smfmullem4  41483  smflimsupmpt  41517
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