Theorem nfabg 2899

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2371. See nfab 2898 for a version with more disjoint variable conditions, but not requiring ax-13 2371. (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 2721	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
3	2	nfci 2880	1 ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1783  {cab 2708  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-nfc 2879

This theorem is referenced by:  nfaba1g  2902  nfiung  4992  nfiing  4993