Theorem nfabd 2923

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfabdw 2922 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2129 and ax-ext 2711. (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 1921	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2718	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		nfabd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfabd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	nfsbd 2530	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
6	2, 5	nfxfrd 1861	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfcd 2894	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:   → wi 4  Ⅎwnf 1790  [wsb 2073   ∈ wcel 2119  {cab 2717  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-nfc 2888

This theorem is referenced by:  nfabd2  2924  nfsbcd  3747  nfcsbd  3856  nfiotad  6446  nfiundg  50165