Theorem hbab1 2749

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 2748	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
2	1	nf5ri 2230	1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1558   ∈ wcel 2142  {cab 2740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741

This theorem is referenced by:  bnj1317  35116  bnj1318  35320  bj-nfsab1  37301