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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 4530. 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 4532 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}) | |
2 | dfif3.1 | . . . . . 6 ⊢ 𝐶 = {𝑥 ∣ 𝜑} | |
3 | biidd 262 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
4 | 3 | cbvabv 2806 | . . . . . 6 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑} |
5 | 2, 4 | eqtri 2761 | . . . . 5 ⊢ 𝐶 = {𝑦 ∣ 𝜑} |
6 | 5 | ineq2i 4210 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∩ {𝑦 ∣ 𝜑}) |
7 | dfrab3 4310 | . . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑦 ∣ 𝜑}) | |
8 | 6, 7 | eqtr4i 2764 | . . 3 ⊢ (𝐴 ∩ 𝐶) = {𝑦 ∈ 𝐴 ∣ 𝜑} |
9 | dfrab3 4310 | . . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} = (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) | |
10 | biidd 262 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜑)) | |
11 | 10 | notabw 4304 | . . . . . 6 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ {𝑧 ∣ 𝜑}) |
12 | biidd 262 | . . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜑)) | |
13 | 12 | cbvabv 2806 | . . . . . . . 8 ⊢ {𝑥 ∣ 𝜑} = {𝑧 ∣ 𝜑} |
14 | 2, 13 | eqtri 2761 | . . . . . . 7 ⊢ 𝐶 = {𝑧 ∣ 𝜑} |
15 | 14 | difeq2i 4120 | . . . . . 6 ⊢ (V ∖ 𝐶) = (V ∖ {𝑧 ∣ 𝜑}) |
16 | 11, 15 | eqtr4i 2764 | . . . . 5 ⊢ {𝑦 ∣ ¬ 𝜑} = (V ∖ 𝐶) |
17 | 16 | ineq2i 4210 | . . . 4 ⊢ (𝐵 ∩ {𝑦 ∣ ¬ 𝜑}) = (𝐵 ∩ (V ∖ 𝐶)) |
18 | 9, 17 | eqtr2i 2762 | . . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = {𝑦 ∈ 𝐵 ∣ ¬ 𝜑} |
19 | 8, 18 | uneq12i 4162 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ({𝑦 ∈ 𝐴 ∣ 𝜑} ∪ {𝑦 ∈ 𝐵 ∣ ¬ 𝜑}) |
20 | 1, 19 | eqtr4i 2764 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 {cab 2710 {crab 3433 Vcvv 3475 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ifcif 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-if 4530 |
This theorem is referenced by: dfif4 4544 |
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