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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 47097 what dfiota4 6553 is to df-iota 6514. operation using the if operator. It is simpler than df-aiota 47097 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 47103 | . 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | |
| 2 | aiotavb 47102 | . . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) | |
| 3 | 2 | biimpi 216 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V) |
| 4 | ifval 4568 | . 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 1540 ∃!weu 2568 {cab 2714 Vcvv 3480 ifcif 4525 ∩ cint 4946 ℩'caiota 47095 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-uni 4908 df-int 4947 df-iota 6514 df-aiota 47097 |
| This theorem is referenced by: (None) |
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