| 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 3458 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | 1 | simprbi 496 | 1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 {crab 3436 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 |
| This theorem is referenced by: infnsuprnmpt 45257 preimagelt 46714 preimalegt 46715 pimrecltpos 46723 pimiooltgt 46725 pimrecltneg 46739 smfaddlem1 46778 smflimlem2 46787 smfrec 46804 smfmullem4 46809 smfdiv 46812 smfsupxr 46831 smfinflem 46832 smflimsuplem7 46841 smflimsuplem8 46842 fsupdm 46857 finfdm 46861 |
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