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| Mirrors > Home > MPE Home > Th. List > ifcld | Structured version Visualization version GIF version | ||
| Description: Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.) |
| Ref | Expression |
|---|---|
| ifcld.a | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| ifcld.b | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
| Ref | Expression |
|---|---|
| ifcld | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ifcld.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
| 2 | ifcld.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
| 3 | ifcl 4571 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) |
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