Theorem nfcrii 2899

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2140, ax-11 2156. (Revised by GG, 23-May-2024.)

Hypothesis

Ref	Expression
nfcrii.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

Step	Hyp	Ref	Expression
1		nfcrii.1	. . 3 ⊢ Ⅎ𝑥𝐴
2	1	nfcri 2896	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nf5ri 2194	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1537   ∈ wcel 2107  Ⅎwnfc 2889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-clel 2815  df-nfc 2891

This theorem is referenced by:  bnj1230  34817  bnj1000  34956  bnj1204  35027  bnj1307  35038  bnj1311  35039  bnj1398  35049  bnj1466  35068  bnj1467  35069  bnj1523  35086