Theorem nfabg 2931

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2403. See nfab 2930 for a version with more disjoint variable conditions, but not requiring ax-13 2403. (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 2753	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
3	2	nfci 2912	1 ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1803  {cab 2740  Ⅎwnfc 2909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-nfc 2911

This theorem is referenced by:  nfaba1g  2933  nfiung  4983  nfiing  4984