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| Description: Equality of class abstractions restricted to a singleton. (Contributed by AV, 17-May-2025.) | 
| Ref | Expression | 
|---|---|
| rabsneq | ⊢ (𝑁 ∈ 𝑉 → {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥 ∈ 𝑉 ∣ (𝑥 = 𝑁 ∧ 𝜓)}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | velsn 4641 | . . . . . 6 ⊢ (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁) | |
| 2 | eleq1a 2835 | . . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → (𝑥 = 𝑁 → 𝑥 ∈ 𝑉)) | |
| 3 | 2 | pm4.71rd 562 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (𝑥 = 𝑁 ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁))) | 
| 4 | 1, 3 | bitrid 283 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑥 ∈ {𝑁} ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁))) | 
| 5 | 4 | anbi1d 631 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → ((𝑥 ∈ {𝑁} ∧ 𝜓) ↔ ((𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁) ∧ 𝜓))) | 
| 6 | anass 468 | . . . 4 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑥 = 𝑁) ∧ 𝜓) ↔ (𝑥 ∈ 𝑉 ∧ (𝑥 = 𝑁 ∧ 𝜓))) | |
| 7 | 5, 6 | bitrdi 287 | . . 3 ⊢ (𝑁 ∈ 𝑉 → ((𝑥 ∈ {𝑁} ∧ 𝜓) ↔ (𝑥 ∈ 𝑉 ∧ (𝑥 = 𝑁 ∧ 𝜓)))) | 
| 8 | 7 | abbidv 2807 | . 2 ⊢ (𝑁 ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ {𝑁} ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝑥 = 𝑁 ∧ 𝜓))}) | 
| 9 | df-rab 3436 | . 2 ⊢ {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ {𝑁} ∧ 𝜓)} | |
| 10 | df-rab 3436 | . 2 ⊢ {𝑥 ∈ 𝑉 ∣ (𝑥 = 𝑁 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ (𝑥 = 𝑁 ∧ 𝜓))} | |
| 11 | 8, 9, 10 | 3eqtr4g 2801 | 1 ⊢ (𝑁 ∈ 𝑉 → {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥 ∈ 𝑉 ∣ (𝑥 = 𝑁 ∧ 𝜓)}) | 
| 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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