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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 4426. 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 4439 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) |
3 | undir 4203 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) | |
4 | undi 4201 | . . . 4 ⊢ (𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) | |
5 | undi 4201 | . . . . 5 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) | |
6 | uncom 4080 | . . . . . 6 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶) | |
7 | unvdif 4381 | . . . . . 6 ⊢ (𝐶 ∪ (V ∖ 𝐶)) = V | |
8 | 6, 7 | ineq12i 4137 | . . . . 5 ⊢ ((𝐶 ∪ 𝐵) ∩ (𝐶 ∪ (V ∖ 𝐶))) = ((𝐵 ∪ 𝐶) ∩ V) |
9 | inv1 4302 | . . . . 5 ⊢ ((𝐵 ∪ 𝐶) ∩ V) = (𝐵 ∪ 𝐶) | |
10 | 5, 8, 9 | 3eqtri 2825 | . . . 4 ⊢ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶))) = (𝐵 ∪ 𝐶) |
11 | 4, 10 | ineq12i 4137 | . . 3 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶)) |
12 | inass 4146 | . . 3 ⊢ (((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ (V ∖ 𝐶))) ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) | |
13 | 11, 12 | eqtri 2821 | . 2 ⊢ ((𝐴 ∪ (𝐵 ∩ (V ∖ 𝐶))) ∩ (𝐶 ∪ (𝐵 ∩ (V ∖ 𝐶)))) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) |
14 | 2, 3, 13 | 3eqtri 2825 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 {cab 2776 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ifcif 4425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 |
This theorem is referenced by: dfif5 4441 |
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