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Mirrors > Home > MPE Home > Th. List > rabbi2dva | Structured version Visualization version GIF version |
Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.) |
Ref | Expression |
---|---|
rabbi2dva.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) |
Ref | Expression |
---|---|
rabbi2dva | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3889 | . 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} | |
2 | rabbi2dva.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) | |
3 | 2 | rabbidva 3425 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
4 | 1, 3 | syl5eq 2845 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ∩ cin 3880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-rab 3115 df-in 3888 |
This theorem is referenced by: fndmdif 6789 bitsshft 15814 sylow3lem2 18745 leordtvallem1 21815 leordtvallem2 21816 ordtt1 21984 xkoccn 22224 txcnmpt 22229 xkopt 22260 ordthmeolem 22406 qustgphaus 22728 itg2monolem1 24354 lhop1 24617 efopn 25249 dirith 26113 pjvec 29479 pjocvec 29480 neibastop3 33823 diarnN 38425 |
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