Theorem nfcii 2911

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 2910	1 ⊢ Ⅎ𝑥𝐴

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   ∈ wcel 2145  Ⅎwnfc 2907

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 2909

This theorem is used by:  bnj1316  35329  bnj1385  35341  bnj1400  35344  bnj1468  35355  bnj1534  35362  bnj1542  35366  bnj1228  35520  bnj1307  35532  bnj1448  35556  bnj1466  35562  bnj1463  35564  bnj1491  35566  bnj1312  35567  bnj1498  35570  bnj1520  35575  bnj1525  35578  bnj1529  35579  bnj1523  35580