Theorem nfsab 2614

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfsab.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfsab	⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Distinct variable group:   𝑥,𝑧

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

Proof of Theorem nfsab

Step	Hyp	Ref	Expression
1		nfsab.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nf5ri 2065	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbab 2613	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
4	3	nf5i 2024	1 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1708   ∈ wcel 1990  {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609

This theorem is referenced by:  nfab  2769  upbdrech  39519  ssfiunibd  39523