Theorem nfsab 2755

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2406. See nfsabg 2756 for a less restrictive version requiring more axioms. (Revised by GG, 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 2234	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbab 2753	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
4	3	nf5i 2184	1 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

This proof depends on syntax axioms:  Ⅎwnf 1816   ∈ wcel 2146  {cab 2743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744

This theorem is used by:  nfab  2933  oaun3lem1  44161  upbdrech  46084  ssfiunibd  46088