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Theorem nfci 2184
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 2183 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfci.1 . 2 𝑥 𝑦𝐴
31, 2mpgbir 1358 1 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wnf 1365  wcel 1409  wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354
This theorem depends on definitions:  df-bi 114  df-nfc 2183
This theorem is referenced by:  nfcii  2185  nfcv  2194  nfab1  2196  nfab  2198
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