Theorem nfci 2754

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 2753	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
2		nfci.1	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	1, 2	mpgbir 1726	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722

This theorem depends on definitions:  df-bi 197  df-nfc 2753

This theorem is referenced by:  nfcii  2755  nfcv  2764  nfab1  2766  nfab  2769  fpwrelmap  29508  esumfzf  30131  bj-nfab1  32785  fsumiunss  39807  climsuse  39840  climinff  39843  fnlimfvre  39906  limsupre3uzlem  39967  pimdecfgtioc  40925  pimincfltioc  40926  smfmullem4  41001  smflimsupmpt  41035