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Theorem nfci 2168
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 2167 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfci.1 . 2 𝑥 𝑦𝐴
31, 2mpgbir 1342 1 𝑥𝐴
Colors of variables: wff set class
Syntax hints:  wnf 1349  wcel 1393  wnfc 2165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338
This theorem depends on definitions:  df-bi 110  df-nfc 2167
This theorem is referenced by:  nfcii  2169  nfcv  2178  nfab1  2180  nfab  2182
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