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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 4037. 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 4050 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) |
3 | undir 3835 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) | |
4 | undi 3833 | . . . 4 ⊢ (𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) | |
5 | undi 3833 | . . . . 5 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) | |
6 | uncom 3719 | . . . . . 6 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶) | |
7 | unvdif 3994 | . . . . . 6 ⊢ (𝐶 ∪ (V ∖ 𝐶)) = V | |
8 | 6, 7 | ineq12i 3774 | . . . . 5 ⊢ ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) = ((𝐵 ∪ 𝐶) ∩ V) |
9 | inv1 3922 | . . . . 5 ⊢ ((𝐵 ∪ 𝐶) ∩ V) = (𝐵 ∪ 𝐶) | |
10 | 5, 8, 9 | 3eqtri 2636 | . . . 4 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = (𝐵 ∪ 𝐶) |
11 | 4, 10 | ineq12i 3774 | . . 3 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶)) |
12 | inass 3785 | . . 3 ⊢ (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) | |
13 | 11, 12 | eqtri 2632 | . 2 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) |
14 | 2, 3, 13 | 3eqtri 2636 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 {cab 2596 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ifcif 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 |
This theorem is referenced by: dfif5 4052 |
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