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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 2137, ax-11 2154 and ax-12 2171. (Revised by Gino Giotto, 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 3073 | . . . . . 6 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | eleq2i 2830 | . . . . 5 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ 𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
3 | df-clab 2716 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑥 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜑)) | |
4 | eleq1w 2821 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
5 | rabrabi.1 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) | |
6 | 5 | bicomd 222 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
7 | 6 | equcoms 2023 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜒)) |
8 | 4, 7 | anbi12d 631 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
9 | 8 | sbievw 2095 | . . . . . 6 ⊢ ([𝑥 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) |
10 | 3, 9 | bitri 274 | . . . . 5 ⊢ (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) |
11 | 2, 10 | bitri 274 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) |
12 | 11 | anbi1i 624 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜒) ∧ 𝜓)) |
13 | anass 469 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜒) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜒 ∧ 𝜓))) | |
14 | 12, 13 | bitri 274 | . 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜒 ∧ 𝜓))) |
15 | 14 | rabbia2 3411 | 1 ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 [wsb 2067 ∈ wcel 2106 {cab 2715 {crab 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 |
This theorem is referenced by: wlksnwwlknvbij 28270 |
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