Theorem nfabg 2926

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfab 2925 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfabg.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfabg	⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}

Proof of Theorem nfabg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfabg.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nfsabg 2749	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
3	2	nfci 2902	1 ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1785  {cab 2735  Ⅎwnfc 2899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-nfc 2901

This theorem is referenced by:  nfaba1g  2928  nfiung  4915  nfiing  4916