Theorem nfcrii 2894

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2147, ax-11 2163. (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 2891	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nf5ri 2203	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 2114  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-clel 2812  df-nfc 2886

This theorem is referenced by:  bnj1230  34978  bnj1000  35117  bnj1204  35188  bnj1307  35199  bnj1311  35200  bnj1398  35210  bnj1466  35229  bnj1467  35230  bnj1523  35247