Theorem nfabg 2913

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2372. See nfab 2912 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (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 2729	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
3	2	nfci 2889	1 ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1787  {cab 2715  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-nfc 2888

This theorem is referenced by:  nfaba1g  2915  nfiung  4953  nfiing  4954