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Mirrors > Home > ILE Home > Th. List > dfif3 | GIF version |
Description: Alternate definition of the conditional operator df-if 3332. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
dfif3.1 | ⊢ 𝐶 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
dfif3 | ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfif6 3333 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}) | |
2 | dfif3.1 | . . . . . 6 ⊢ 𝐶 = {𝑥 ∣ 𝜑} | |
3 | biidd 161 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
4 | 3 | cbvabv 2161 | . . . . . 6 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑} |
5 | 2, 4 | eqtri 2060 | . . . . 5 ⊢ 𝐶 = {𝑦 ∣ 𝜑} |
6 | 5 | ineq2i 3135 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∩ {𝑦 ∣ 𝜑}) |
7 | dfrab3 3213 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑦 ∣ 𝜑}) | |
8 | 6, 7 | eqtr4i 2063 | . . 3 ⊢ (𝐴 ∩ 𝐶) = {𝑦 ∈ 𝐴 ∣ 𝜑} |
9 | dfrab3 3213 | . . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) | |
10 | notab 3207 | . . . . . 6 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜑}) | |
11 | 5 | difeq2i 3059 | . . . . . 6 ⊢ (V ∖ 𝐶) = (V ∖ {𝑦 ∣ 𝜑}) |
12 | 10, 11 | eqtr4i 2063 | . . . . 5 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶) |
13 | 12 | ineq2i 3135 | . . . 4 ⊢ (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶)) |
14 | 9, 13 | eqtr2i 2061 | . . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} |
15 | 8, 14 | uneq12i 3095 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}) |
16 | 1, 15 | eqtr4i 2063 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1243 {cab 2026 {crab 2310 Vcvv 2557 ∖ cdif 2914 ∪ cun 2915 ∩ cin 2916 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-in1 544 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-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 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-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-if 3332 |
This theorem is referenced by: (None) |
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