Theorem nfci 2921

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	⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Assertion

Ref	Expression
nfci	⊢ Ⅎ𝑥𝐴

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfci

Step	Hyp	Ref	Expression
1		df-nfc 2920	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
2		nfci.1	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	1, 2	mpgbir 1843	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:  Ⅎwnf 1827   ∈ wcel 2106  Ⅎwnfc 2918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839

This theorem depends on definitions:  df-bi 199  df-nfc 2920

This theorem is referenced by:  nfcii  2922  nfcv  2933  nfab1  2935  nfab  2939  fpwrelmap  30074  esumfzf  30729  bj-nfab1  33362  fsumiunss  40708  climsuse  40741  climinff  40744  fnlimfvre  40807  limsupre3uzlem  40868  pimdecfgtioc  41845  pimincfltioc  41846  smfmullem4  41921  smflimsupmpt  41955