Theorem rabsneq 4601

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 4598	. . . . 5 ⊢ (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
2		eleq1a 2857	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (𝑥 = 𝑁 → 𝑥 ∈ 𝑉))
3	2	pm4.71rd 570	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑥 = 𝑁 ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁)))
4	1, 3	bitrid 285	. . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑥 ∈ {𝑁} ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁)))
5	4	anbi1d 640	. . 3 ⊢ (𝑁 ∈ 𝑉 → ((𝑥 ∈ {𝑁} ∧ 𝜓) ↔ ((𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁) ∧ 𝜓)))
6		anass 472	. . 3 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁) ∧ 𝜓) ↔ (𝑥 ∈ 𝑉 ∧ (𝑥 = 𝑁 ∧ 𝜓)))
7	5, 6	bitrdi 289	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝑥 ∈ {𝑁} ∧ 𝜓) ↔ (𝑥 ∈ 𝑉 ∧ (𝑥 = 𝑁 ∧ 𝜓))))
8	7	rabbidva2 3416	1 ⊢ (𝑁 ∈ 𝑉 → {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥 ∈ 𝑉 ∣ (𝑥 = 𝑁 ∧ 𝜓)})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {crab 3414  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-sn 4583

This theorem is referenced by:  dfsclnbgr6  48480