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Mirrors > Home > MPE Home > Th. List > dfif3 | 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.) (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 4428 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}) | |
2 | dfif3.1 | . . . . . 6 ⊢ 𝐶 = {𝑥 ∣ 𝜑} | |
3 | biidd 265 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
4 | 3 | cbvabv 2866 | . . . . . 6 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑} |
5 | 2, 4 | eqtri 2821 | . . . . 5 ⊢ 𝐶 = {𝑦 ∣ 𝜑} |
6 | 5 | ineq2i 4136 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∩ {𝑦 ∣ 𝜑}) |
7 | dfrab3 4230 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑦 ∣ 𝜑}) | |
8 | 6, 7 | eqtr4i 2824 | . . 3 ⊢ (𝐴 ∩ 𝐶) = {𝑦 ∈ 𝐴 ∣ 𝜑} |
9 | dfrab3 4230 | . . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) | |
10 | notab 4225 | . . . . . 6 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜑}) | |
11 | 5 | difeq2i 4047 | . . . . . 6 ⊢ (V ∖ 𝐶) = (V ∖ {𝑦 ∣ 𝜑}) |
12 | 10, 11 | eqtr4i 2824 | . . . . 5 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶) |
13 | 12 | ineq2i 4136 | . . . 4 ⊢ (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶)) |
14 | 9, 13 | eqtr2i 2822 | . . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} |
15 | 8, 14 | uneq12i 4088 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}) |
16 | 1, 15 | eqtr4i 2824 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 {cab 2776 {crab 3110 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-if 4426 |
This theorem is referenced by: dfif4 4440 |
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