Theorem nfcrii 2887

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2129, ax-11 2146. (Revised by Gino Giotto, 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 2882	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nf5ri 2180	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1531   ∈ wcel 2098  Ⅎwnfc 2875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-clel 2802  df-nfc 2877

This theorem is referenced by:  nfcriOLDOLDOLD  2889  bnj1230  34304  bnj1000  34443  bnj1204  34514  bnj1307  34525  bnj1311  34526  bnj1398  34536  bnj1466  34555  bnj1467  34556  bnj1523  34573