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  34786  bnj1385  34798  bnj1400  34801  bnj1468  34812  bnj1534  34819  bnj1542  34823  bnj1228  34977  bnj1307  34989  bnj1448  35013  bnj1466  35019  bnj1463  35021  bnj1491  35023  bnj1312  35024  bnj1498  35027  bnj1520  35032  bnj1525  35035  bnj1529  35036  bnj1523  35037