Theorem nfcrii 2899

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2137, ax-11 2154. (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 2894	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nf5ri 2188	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1537   ∈ wcel 2106  Ⅎwnfc 2887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787  df-clel 2816  df-nfc 2889

This theorem is referenced by:  nfcriOLDOLDOLD  2901  bnj1230  32782  bnj1000  32921  bnj1204  32992  bnj1307  33003  bnj1311  33004  bnj1398  33014  bnj1466  33033  bnj1467  33034  bnj1523  33051