Theorem hbab1 2767

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 2760	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
2		hbs1 2203	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3	1, 2	hbxfrbi 1787	1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1505  [wsb 2015   ∈ wcel 2050  {cab 2759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760

This theorem is referenced by:  nfsab1OLD  2769  abeq2  2898  abbi  2908  abeq2fOLD  2966  bnj1317  31738  bnj1318  31939