Theorem nfcii 2881

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	nf5i 2147	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nfci 2880	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:   → wi 4  ∀wal 1538   ∈ wcel 2109  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-10 2142

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784  df-nfc 2879

This theorem is referenced by:  bnj1316  34817  bnj1385  34829  bnj1400  34832  bnj1468  34843  bnj1534  34850  bnj1542  34854  bnj1228  35008  bnj1307  35020  bnj1448  35044  bnj1466  35050  bnj1463  35052  bnj1491  35054  bnj1312  35055  bnj1498  35058  bnj1520  35063  bnj1525  35066  bnj1529  35067  bnj1523  35068