Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabidim2 | Structured version Visualization version GIF version |
Description: Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rabidim2 | ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabid 3380 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | simprbi 499 | 1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 {crab 3144 |
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 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-rab 3149 |
This theorem is referenced by: infnsuprnmpt 41529 preimagelt 42987 preimalegt 42988 pimrecltpos 42994 pimiooltgt 42996 pimrecltneg 43008 smfaddlem1 43046 smflimlem2 43055 smfrec 43071 smfmullem4 43076 smfdiv 43079 smfsupxr 43097 smfinflem 43098 smflimsuplem7 43107 smflimsuplem8 43108 |
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