Theorem nfcrii 2917

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2177, ax-11 2193. (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 2914	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	2	nf5ri 2231	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   ∈ wcel 2144  Ⅎwnfc 2907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-clel 2835  df-nfc 2909

This theorem is used by:  nfralw  3309  bnj1230  35482  bnj1000  35621  bnj1204  35692  bnj1307  35703  bnj1311  35704  bnj1398  35714  bnj1466  35733  bnj1467  35734  bnj1523  35751