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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 2079 and ax-11 2093. (Revised by Gino Giotto, 20-Aug-2023.) |
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 | rabrabi.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜑)) | |
2 | 1 | cbvrabv 3412 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑦 ∈ 𝐴 ∣ 𝜑} |
3 | 2 | rabeqi 3405 | . 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ∣ 𝜓} = {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} |
4 | rabrab 3318 | . 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} | |
5 | 3, 4 | eqtr3i 2804 | 1 ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 {crab 3092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-rab 3097 |
This theorem is referenced by: wlksnwwlknvbij 27408 |
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