Theorem nfcrii 2890

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2130, ax-11 2147. (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 2885	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nf5ri 2181	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1532   ∈ wcel 2099  Ⅎwnfc 2878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-12 2164

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779  df-clel 2805  df-nfc 2880

This theorem is referenced by:  nfcriOLDOLDOLD  2892  bnj1230  34369  bnj1000  34508  bnj1204  34579  bnj1307  34590  bnj1311  34591  bnj1398  34601  bnj1466  34620  bnj1467  34621  bnj1523  34638