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Mirrors > Home > ILE Home > Th. List > dfif6 | GIF version |
Description: An alternate definition of the conditional operator df-if 3422 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
dfif6 | ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unab 3290 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
2 | df-rab 2384 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
3 | df-rab 2384 | . . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)} | |
4 | 2, 3 | uneq12i 3175 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∪ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)}) |
5 | df-if 3422 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
6 | 1, 4, 5 | 3eqtr4ri 2131 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ∨ wo 670 = wceq 1299 ∈ wcel 1448 {cab 2086 {crab 2379 ∪ cun 3019 ifcif 3421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rab 2384 df-v 2643 df-un 3025 df-if 3422 |
This theorem is referenced by: ifeq1 3424 ifeq2 3425 dfif3 3434 |
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