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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 3336 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | adantl 262 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴) |
3 | ifcldcd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
4 | 3 | adantr 261 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶) |
5 | 2, 4 | eqeltrd 2114 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
6 | iffalse 3339 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
7 | 6 | adantl 262 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵) |
8 | ifcldcd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
9 | 8 | adantr 261 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶) |
10 | 7, 9 | eqeltrd 2114 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
11 | ifcldcd.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
12 | df-dc 743 | . . 3 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓)) | |
13 | 11, 12 | sylib 127 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
14 | 5, 10, 13 | mpjaodan 711 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∨ wo 629 DECID wdc 742 = wceq 1243 ∈ wcel 1393 ifcif 3331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-if 3332 |
This theorem is referenced by: uzin2 9586 |
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