Theorem nfsab 2345

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfsab.1	⊢ Ⅎxφ

Assertion

Ref	Expression
nfsab	⊢ Ⅎx z ∈ {y ∣ φ}

Distinct variable group:   x,z

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem nfsab

Step	Hyp	Ref	Expression
1		nfsab.1	. . . 4 ⊢ Ⅎxφ
2	1	nfri 1762	. . 3 ⊢ (φ → ∀xφ)
3	2	hbab 2344	. 2 ⊢ (z ∈ {y ∣ φ} → ∀x z ∈ {y ∣ φ})
4	3	nfi 1551	1 ⊢ Ⅎx z ∈ {y ∣ φ}

Syntax hints:  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340

This theorem is referenced by:  nfab  2493