Theorem nfabd 2977

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfabdw 2976 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2121 and ax-ext 2770. (Revised by Wolf Lammen, 23-May-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfabd.1	⊢ Ⅎ𝑦𝜑
nfabd.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfabd	⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Proof of Theorem nfabd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2777	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		nfabd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfabd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	nfsbd 2541	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
6	2, 5	nfxfrd 1855	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfcd 2944	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-nfc 2938

This theorem is referenced by:  nfabd2  2978  nfsbcd  3744  nfcsbd  3853  nfiotad  6288  nfiundg  45205