Theorem nfabd 2948

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker nfabdw 2947 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2157 and ax-ext 2735. (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 1949	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2742	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		nfabd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfabd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	nfsbd 2554	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
6	2, 5	nfxfrd 1887	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfcd 2919	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1816  [wsb 2101   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-nfc 2913

This theorem is used by:  nfabd2  2949  nfsbcd  3766  nfcsbd  3876  nfiotad  6525  nfiundg  51189