Theorem nfsabg 2751

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2401. See nfsab 2750 for a version with more disjoint variable conditions, but not requiring ax-13 2401. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfsabg.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfsabg	⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Distinct variable group:   𝑥,𝑧

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

Proof of Theorem nfsabg

Step	Hyp	Ref	Expression
1		nfsabg.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nf5ri 2231	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbabg 2749	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
4	3	nf5i 2182	1 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

This proof depends on syntax axioms:  Ⅎwnf 1815   ∈ wcel 2144  {cab 2738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739

This theorem is used by:  nfabg  2929