Theorem nfcii 2884

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 2151	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nfci 2883	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2113  Ⅎwnfc 2880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-10 2146

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-nf 1785  df-nfc 2882

This theorem is referenced by:  bnj1316  34855  bnj1385  34867  bnj1400  34870  bnj1468  34881  bnj1534  34888  bnj1542  34892  bnj1228  35046  bnj1307  35058  bnj1448  35082  bnj1466  35088  bnj1463  35090  bnj1491  35092  bnj1312  35093  bnj1498  35096  bnj1520  35101  bnj1525  35104  bnj1529  35105  bnj1523  35106