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Theorem nfci 2751
 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 2750 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfci.1 . 2 𝑥 𝑦𝐴
31, 2mpgbir 1723 1 𝑥𝐴
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnf 1705   ∈ wcel 1987  Ⅎwnfc 2748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719 This theorem depends on definitions:  df-bi 197  df-nfc 2750 This theorem is referenced by:  nfcii  2752  nfcv  2761  nfab1  2763  nfab  2765  fpwrelmap  29392  esumfzf  29954  bj-nfab1  32481  fsumiunss  39243  climsuse  39276  climinff  39279  fnlimfvre  39342  limsupre3uzlem  39403  pimdecfgtioc  40262  pimincfltioc  40263  smfmullem4  40338  smflimsupmpt  40372
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