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Theorem nfci 2783
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 2782 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
2 nfci.1 . 2 𝑥 𝑦𝐴
31, 2mpgbir 1766 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wnf 1748  wcel 2030  wnfc 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762
This theorem depends on definitions:  df-bi 197  df-nfc 2782
This theorem is referenced by:  nfcii  2784  nfcv  2793  nfab1  2795  nfab  2798  fpwrelmap  29636  esumfzf  30259  bj-nfab1  32910  fsumiunss  40125  climsuse  40158  climinff  40161  fnlimfvre  40224  limsupre3uzlem  40285  pimdecfgtioc  41246  pimincfltioc  41247  smfmullem4  41322  smflimsupmpt  41356
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