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Theorem rabsneq 4643
Description: Equality of class abstractions restricted to a singleton. (Contributed by AV, 17-May-2025.)
Assertion
Ref Expression
rabsneq (𝑁𝑉 → {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥𝑉 ∣ (𝑥 = 𝑁𝜓)})
Distinct variable groups:   𝑥,𝑁   𝑥,𝑉
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabsneq
StepHypRef Expression
1 velsn 4641 . . . . . 6 (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
2 eleq1a 2835 . . . . . . 7 (𝑁𝑉 → (𝑥 = 𝑁𝑥𝑉))
32pm4.71rd 562 . . . . . 6 (𝑁𝑉 → (𝑥 = 𝑁 ↔ (𝑥𝑉𝑥 = 𝑁)))
41, 3bitrid 283 . . . . 5 (𝑁𝑉 → (𝑥 ∈ {𝑁} ↔ (𝑥𝑉𝑥 = 𝑁)))
54anbi1d 631 . . . 4 (𝑁𝑉 → ((𝑥 ∈ {𝑁} ∧ 𝜓) ↔ ((𝑥𝑉𝑥 = 𝑁) ∧ 𝜓)))
6 anass 468 . . . 4 (((𝑥𝑉𝑥 = 𝑁) ∧ 𝜓) ↔ (𝑥𝑉 ∧ (𝑥 = 𝑁𝜓)))
75, 6bitrdi 287 . . 3 (𝑁𝑉 → ((𝑥 ∈ {𝑁} ∧ 𝜓) ↔ (𝑥𝑉 ∧ (𝑥 = 𝑁𝜓))))
87abbidv 2807 . 2 (𝑁𝑉 → {𝑥 ∣ (𝑥 ∈ {𝑁} ∧ 𝜓)} = {𝑥 ∣ (𝑥𝑉 ∧ (𝑥 = 𝑁𝜓))})
9 df-rab 3436 . 2 {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ {𝑁} ∧ 𝜓)}
10 df-rab 3436 . 2 {𝑥𝑉 ∣ (𝑥 = 𝑁𝜓)} = {𝑥 ∣ (𝑥𝑉 ∧ (𝑥 = 𝑁𝜓))}
118, 9, 103eqtr4g 2801 1 (𝑁𝑉 → {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥𝑉 ∣ (𝑥 = 𝑁𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {cab 2713  {crab 3435  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-sn 4626
This theorem is referenced by:  dfsclnbgr6  47849
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