Theorem hbab 2072

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)

Hypothesis

Ref	Expression
hbab.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

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

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbab

Step	Hyp	Ref	Expression
1		df-clab 2068	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑)
2		hbab.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbsb 1864	. 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑)
4	1, 3	hbxfrbi 1401	1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})

Syntax hints:   → wi 4  ∀wal 1282   ∈ wcel 1433  [wsb 1685  {cab 2067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068

This theorem is referenced by:  nfsab  2073