Theorem nfsabg 2731

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2380. See nfsab 2730 for a version with more disjoint variable conditions, but not requiring ax-13 2380. (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 2196	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbabg 2729	. 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
4	3	nf5i 2146	1 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1781   ∈ wcel 2108  {cab 2717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718

This theorem is referenced by:  nfabg  2915