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Mirrors > Home > ILE Home > Th. List > ifelpwund | GIF version |
Description: Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.) |
Ref | Expression |
---|---|
ifelpwund.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ifelpwund.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
ifelpwund | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifelpwund.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | ifelpwund.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | ifelpwung 4466 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 ∪ cun 3119 ifcif 3526 𝒫 cpw 3566 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 |
This theorem is referenced by: ifexd 4469 |
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