Theorem nfabd 2949

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker nfabdw 2948 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2156 and ax-ext 2737. (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 1947	. 2 ⊢ Ⅎ𝑧𝜑
2		df-clab 2744	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
3		nfabd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfabd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	nfsbd 2556	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
6	2, 5	nfxfrd 1887	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfcd 2920	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1816  [wsb 2099   ∈ wcel 2146  {cab 2743  Ⅎwnfc 2912

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 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406

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 2100  df-clab 2744  df-nfc 2914

This theorem is used by:  nfabd2  2950  nfsbcd  3770  nfcsbd  3879  nfiotad  6501  nfiundg  50486