Theorem nfcii 2880

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 2879	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:   → wi 4  ∀wal 1538   ∈ wcel 2109  Ⅎwnfc 2876

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 2878

This theorem is referenced by:  bnj1316  34810  bnj1385  34822  bnj1400  34825  bnj1468  34836  bnj1534  34843  bnj1542  34847  bnj1228  35001  bnj1307  35013  bnj1448  35037  bnj1466  35043  bnj1463  35045  bnj1491  35047  bnj1312  35048  bnj1498  35051  bnj1520  35056  bnj1525  35059  bnj1529  35060  bnj1523  35061