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| Mirrors > Home > MPE Home > Th. List > rabrabi | Structured version Visualization version GIF version | ||
| Description: Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2141, ax-11 2157 and ax-12 2177. (Revised by GG, 12-Oct-2024.) |
| Ref | Expression |
|---|---|
| rabrabi.1 | ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) |
| Ref | Expression |
|---|---|
| rabrabi | ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-rab 3437 | . . . . . 6 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
| 2 | 1 | eleq2i 2833 | . . . . 5 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
| 3 | df-clab 2715 | . . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑥 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜑)) | |
| 4 | eleq1w 2824 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
| 5 | rabrabi.1 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) | |
| 6 | 5 | bicomd 223 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
| 7 | 6 | equcoms 2019 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜒)) |
| 8 | 4, 7 | anbi12d 632 | . . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
| 9 | 8 | sbievw 2093 | . . . . 5 ⊢ ([𝑥 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) |
| 10 | 2, 3, 9 | 3bitri 297 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) |
| 11 | 10 | anbi1i 624 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜒) ∧ 𝜓)) |
| 12 | anass 468 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜒) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜒 ∧ 𝜓))) | |
| 13 | 11, 12 | bitri 275 | . 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜒 ∧ 𝜓))) |
| 14 | 13 | rabbia2 3439 | 1 ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 [wsb 2064 ∈ wcel 2108 {cab 2714 {crab 3436 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 |
| This theorem is referenced by: wlksnwwlknvbij 29928 |
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