Theorem nfcii 2888

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 2152	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nfci 2887	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 2114  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-10 2147

This theorem depends on definitions:  df-bi 207  df-ex 1782  df-nf 1786  df-nfc 2886

This theorem is referenced by:  bnj1316  34995  bnj1385  35007  bnj1400  35010  bnj1468  35021  bnj1534  35028  bnj1542  35032  bnj1228  35186  bnj1307  35198  bnj1448  35222  bnj1466  35228  bnj1463  35230  bnj1491  35232  bnj1312  35233  bnj1498  35236  bnj1520  35241  bnj1525  35244  bnj1529  35245  bnj1523  35246