Theorem hbab1 2640

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.)

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		df-clab 2638	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2		hbs1 2464	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3	1, 2	hbxfrbi 1792	1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1521  [wsb 1937   ∈ wcel 2030  {cab 2637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638

This theorem is referenced by:  nfsab1  2641  abeq2  2761  abbi  2766  abeq2f  2821  bnj1317  31018  bnj1318  31219