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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 3484 | . . . 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 2217 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
6 | iffalse 3487 | . . . 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 2217 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
11 | ifcldcd.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
12 | df-dc 821 | . . 3 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓)) | |
13 | 11, 12 | sylib 121 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
14 | 5, 10, 13 | mpjaodan 788 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 DECID wdc 820 = wceq 1332 ∈ wcel 1481 ifcif 3479 |
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 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-if 3480 |
This theorem is referenced by: fimax2gtrilemstep 6802 fodjuf 7025 fodjum 7026 fodju0 7027 nnnninf 7031 mkvprop 7040 xaddf 9657 xaddval 9658 uzin2 10791 fsum3ser 11198 fsumsplit 11208 explecnv 11306 ennnfonelemp1 11955 nnsf 13374 peano4nninf 13375 nninfalllemn 13377 nninfsellemcl 13382 nninffeq 13391 dceqnconst 13423 |
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