Theorem nfsab 2815

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2389. See nfsabg 2816 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)

Hypothesis

Ref	Expression
nfsab.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfsab	⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Distinct variable groups:   𝑥,𝑦   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 2194	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbab 2813	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
4	3	nf5i 2149	1 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1783   ∈ wcel 2113  {cab 2802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803

This theorem is referenced by:  nfab  2987  upbdrech  41578  ssfiunibd  41582