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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 2144 and ax-11 2160. (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 3494 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑦 ∈ 𝐴 ∣ 𝜑} |
3 | 2 | rabeqi 3485 | . 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ∣ 𝜓} = {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} |
4 | rabrab 3382 | . 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} | |
5 | 3, 4 | eqtr3i 2849 | 1 ⊢ {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 {crab 3145 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-rab 3150 |
This theorem is referenced by: wlksnwwlknvbij 27690 |
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