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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 3449 | . . . 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 2194 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
6 | iffalse 3452 | . . . 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 2194 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
11 | ifcldcd.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
12 | df-dc 805 | . . 3 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓)) | |
13 | 11, 12 | sylib 121 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
14 | 5, 10, 13 | mpjaodan 772 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 682 DECID wdc 804 = wceq 1316 ∈ wcel 1465 ifcif 3444 |
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 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-if 3445 |
This theorem is referenced by: fimax2gtrilemstep 6762 fodjuf 6985 fodjum 6986 fodju0 6987 nnnninf 6991 mkvprop 7000 xaddf 9595 xaddval 9596 uzin2 10727 fsum3ser 11134 fsumsplit 11144 explecnv 11242 ennnfonelemp1 11846 nnsf 13126 peano4nninf 13127 nninfalllemn 13129 nninfsellemcl 13134 nninffeq 13143 |
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