Theorem nfci 2302

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 2301	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
2		nfci.1	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	1, 2	mpgbir 1446	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:  Ⅎwnf 1453   ∈ wcel 2141  Ⅎwnfc 2299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442

This theorem depends on definitions:  df-bi 116  df-nfc 2301

This theorem is referenced by:  nfcii  2303  nfcv  2312  nfab1  2314  nfab  2317