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  34978  bnj1385  34990  bnj1400  34993  bnj1468  35004  bnj1534  35011  bnj1542  35015  bnj1228  35169  bnj1307  35181  bnj1448  35205  bnj1466  35211  bnj1463  35213  bnj1491  35215  bnj1312  35216  bnj1498  35219  bnj1520  35224  bnj1525  35227  bnj1529  35228  bnj1523  35229