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| Mirrors > Home > MPE Home > Th. List > dfif4 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the conditional operator df-if 4490. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) |
| Ref | Expression |
|---|---|
| dfif3.1 | ⊢ 𝐶 = {𝑥 ∣ 𝜑} |
| Ref | Expression |
|---|---|
| dfif4 | ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfif3.1 | . . 3 ⊢ 𝐶 = {𝑥 ∣ 𝜑} | |
| 2 | 1 | dfif3 4504 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) |
| 3 | undir 4240 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) | |
| 4 | undi 4238 | . . . 4 ⊢ (𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) | |
| 5 | undi 4238 | . . . . 5 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) | |
| 6 | uncom 4112 | . . . . . 6 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶) | |
| 7 | unvdif 4436 | . . . . . 6 ⊢ (𝐶 ∪ (V ∖ 𝐶)) = V | |
| 8 | 6, 7 | ineq12i 4171 | . . . . 5 ⊢ ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) = ((𝐵 ∪ 𝐶) ∩ V) |
| 9 | inv1 4355 | . . . . 5 ⊢ ((𝐵 ∪ 𝐶) ∩ V) = (𝐵 ∪ 𝐶) | |
| 10 | 5, 8, 9 | 3eqtri 2792 | . . . 4 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = (𝐵 ∪ 𝐶) |
| 11 | 4, 10 | ineq12i 4171 | . . 3 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶)) |
| 12 | inass 4180 | . . 3 ⊢ (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) | |
| 13 | 11, 12 | eqtri 2788 | . 2 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) |
| 14 | 2, 3, 13 | 3eqtri 2792 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: = wceq 1570 {cab 2743 Vcvv 3457 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ifcif 4489 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-ext 2737 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2744 df-cleq 2757 df-clel 2840 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 |
| This theorem is used by: dfif5 4506 ifssun 4507 |
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