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Theorem nfci 2741
 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 2740 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfci.1 . 2 𝑥 𝑦𝐴
31, 2mpgbir 1717 1 𝑥𝐴
 Colors of variables: wff setvar class Syntax hints:  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713 This theorem depends on definitions:  df-bi 196  df-nfc 2740 This theorem is referenced by:  nfcii  2742  nfcv  2751  nfab1  2753  nfab  2755  fpwrelmap  28896  esumfzf  29458  bj-nfab1  31973  fsumiunss  38642  climsuse  38675  climinff  38678  fnlimfvre  38741  pimdecfgtioc  39602  pimincfltioc  39603  smfmullem4  39679
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