Theorem rabsneq 4610

Description: Equality of class abstractions restricted to a singleton. (Contributed by AV, 17-May-2025.)

Assertion

Ref	Expression
rabsneq	⊢ (𝑁 ∈ 𝑉 → {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥 ∈ 𝑉 ∣ (𝑥 = 𝑁 ∧ 𝜓)})

Distinct variable groups:   𝑥,𝑁   𝑥,𝑉

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabsneq

Step	Hyp	Ref	Expression
1		velsn 4607	. . . . 5 ⊢ (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
2		eleq1a 2860	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (𝑥 = 𝑁 → 𝑥 ∈ 𝑉))
3	2	pm4.71rd 572	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑥 = 𝑁 ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁)))
4	1, 3	bitrid 286	. . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑥 ∈ {𝑁} ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁)))
5	4	anbi1d 643	. . 3 ⊢ (𝑁 ∈ 𝑉 → ((𝑥 ∈ {𝑁} ∧ 𝜓) ↔ ((𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁) ∧ 𝜓)))
6		anass 474	. . 3 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁) ∧ 𝜓) ↔ (𝑥 ∈ 𝑉 ∧ (𝑥 = 𝑁 ∧ 𝜓)))
7	5, 6	bitrdi 290	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝑥 ∈ {𝑁} ∧ 𝜓) ↔ (𝑥 ∈ 𝑉 ∧ (𝑥 = 𝑁 ∧ 𝜓))))
8	7	rabbidva2 3420	1 ⊢ (𝑁 ∈ 𝑉 → {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥 ∈ 𝑉 ∣ (𝑥 = 𝑁 ∧ 𝜓)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3418  {csn 4591

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-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-sn 4592

This theorem is used by:  dfsclnbgr6  48681