Theorem hbab1 2722

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)

Assertion

Ref	Expression
hbab1	⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem hbab1

Step	Hyp	Ref	Expression
1		nfsab1 2721	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
2	1	nf5ri 2186	1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1537   ∈ wcel 2104  {cab 2713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714

This theorem is referenced by:  abeq2  2870  bnj1317  32850  bnj1318  33054  bj-nfsab1  35047