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Mirrors > Home > ILE Home > Th. List > ifcldcd | GIF version |
Description: Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.) |
Ref | Expression |
---|---|
ifcldcd.a | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
ifcldcd.b | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
ifcldcd.dc | ⊢ (𝜑 → DECID 𝜓) |
Ref | Expression |
---|---|
ifcldcd | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3479 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴) |
3 | ifcldcd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
4 | 3 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) |
5 | 2, 4 | eqeltrd 2216 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
6 | iffalse 3482 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
7 | 6 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵) |
8 | ifcldcd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
9 | 8 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) |
10 | 7, 9 | eqeltrd 2216 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
11 | ifcldcd.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
12 | df-dc 820 | . . 3 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓)) | |
13 | 11, 12 | sylib 121 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
14 | 5, 10, 13 | mpjaodan 787 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 697 DECID wdc 819 = wceq 1331 ∈ wcel 1480 ifcif 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-if 3475 |
This theorem is referenced by: fimax2gtrilemstep 6794 fodjuf 7017 fodjum 7018 fodju0 7019 nnnninf 7023 mkvprop 7032 xaddf 9627 xaddval 9628 uzin2 10759 fsum3ser 11166 fsumsplit 11176 explecnv 11274 ennnfonelemp1 11919 nnsf 13199 peano4nninf 13200 nninfalllemn 13202 nninfsellemcl 13207 nninffeq 13216 |
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