Theorem nfcrii 2222

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfcri.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcrii	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcrii

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcri.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		nfcr 2221	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴)
3	1, 2	ax-mp 7	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4	3	nfri 1458	. 2 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
5	4	hblem 2196	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1288  Ⅎwnf 1395   ∈ wcel 1439  Ⅎwnfc 2216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218

This theorem is referenced by:  nfcri  2223