Theorem nfabg 2938

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2410. See nfab 2937 for a version with more disjoint variable conditions, but not requiring ax-13 2410. (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 2760	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
3	2	nfci 2919	1 ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1810  {cab 2747  Ⅎwnfc 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-nfc 2918

This theorem is referenced by:  nfaba1g  2940  nfiung  4994  nfiing  4995