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Mirrors > Home > MPE Home > Th. List > rabrab | Structured version Visualization version GIF version |
Description: Abstract builder restricted to another restricted abstract builder. (Contributed by Thierry Arnoux, 30-Aug-2017.) |
Ref | Expression |
---|---|
rabrab | ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabid 3373 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | anbi1i 625 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) |
3 | anass 471 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
4 | 2, 3 | bitri 277 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) |
5 | 4 | abbii 2885 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))} |
6 | df-rab 3142 | . 2 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∧ 𝜓)} | |
7 | df-rab 3142 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))} | |
8 | 5, 6, 7 | 3eqtr4i 2853 | 1 ⊢ {𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2798 {crab 3137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-sb 2070 df-clab 2799 df-cleq 2813 df-clel 2891 df-rab 3142 |
This theorem is referenced by: rabrabi 3480 extwwlkfab 28110 fpwrelmapffs 30451 |
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