Theorem nfcrii 2896

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2152, ax-11 2168. (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 2893	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nf5ri 2207	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1545   ∈ wcel 2119  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-clel 2814  df-nfc 2888

This theorem is referenced by:  nfralw  3286  bnj1230  34984  bnj1000  35123  bnj1204  35194  bnj1307  35205  bnj1311  35206  bnj1398  35216  bnj1466  35235  bnj1467  35236  bnj1523  35253