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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfaiota3 | Structured version Visualization version GIF version |
Description: Alternate definition of ℩': this is to df-aiota 44192 what dfiota4 6350 is to df-iota 6316. operation using the if operator. It is simpler than df-aiota 44192 and uses no dummy variables, so it would be the preferred definition if ℩' becomes the description binder used in set.mm. (Contributed by BJ, 31-Aug-2024.) |
Ref | Expression |
---|---|
dfaiota3 | ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aiotaint 44198 | . 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | |
2 | aiotavb 44197 | . . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) | |
3 | 2 | biimpi 219 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V) |
4 | ifval 4467 | . 2 ⊢ ((℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V))) | |
5 | 1, 3, 4 | mpbir2an 711 | 1 ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∃!weu 2567 {cab 2714 Vcvv 3398 ifcif 4425 ∩ cint 4845 ℩'caiota 44190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-uni 4806 df-int 4846 df-iota 6316 df-aiota 44192 |
This theorem is referenced by: (None) |
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