Theorem nfcii 2912

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 2183	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nfci 2911	1 ⊢ Ⅎ𝑥𝐴

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   ∈ wcel 2145  Ⅎwnfc 2908

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-10 2178

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817  df-nfc 2910

This theorem is used by:  bnj1316  35443  bnj1385  35455  bnj1400  35458  bnj1468  35469  bnj1534  35476  bnj1542  35480  bnj1228  35634  bnj1307  35646  bnj1448  35670  bnj1466  35676  bnj1463  35678  bnj1491  35680  bnj1312  35681  bnj1498  35684  bnj1520  35689  bnj1525  35692  bnj1529  35693  bnj1523  35694