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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 45872 what dfiota4 6535 is to df-iota 6495. operation using the if operator. It is simpler than df-aiota 45872 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 45878 | . 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) | |
2 | aiotavb 45877 | . . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V) | |
3 | 2 | biimpi 215 | . 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V) |
4 | ifval 4570 | . 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 1541 ∃!weu 2562 {cab 2709 Vcvv 3474 ifcif 4528 ∩ cint 4950 ℩'caiota 45870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-uni 4909 df-int 4951 df-iota 6495 df-aiota 45872 |
This theorem is referenced by: (None) |
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