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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 46512 what dfiota4 6545 is to df-iota 6505. operation using the if operator. It is simpler than df-aiota 46512 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 46518 | . 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | |
2 | aiotavb 46517 | . . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) | |
3 | 2 | biimpi 215 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V) |
4 | ifval 4574 | . 2 ⊢ ((℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V))) | |
5 | 1, 3, 4 | mpbir2an 709 | 1 ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∃!weu 2557 {cab 2705 Vcvv 3473 ifcif 4532 ∩ cint 4953 ℩'caiota 46510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-uni 4913 df-int 4954 df-iota 6505 df-aiota 46512 |
This theorem is referenced by: (None) |
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