Theorem nfcii 2340

Description: Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfcii.1	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Assertion

Ref	Expression
nfcii	⊢ Ⅎ𝑥𝐴

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfcii

Step	Hyp	Ref	Expression
1		nfcii.1	. . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
2	1	nfi 1486	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nfci 2339	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:   → wi 4  ∀wal 1371   ∈ wcel 2177  Ⅎwnfc 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1473

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-nfc 2338

This theorem is referenced by: (None)