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Mirrors > Home > MPE Home > Th. List > ifnot | Structured version Visualization version GIF version |
Description: Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.) |
Ref | Expression |
---|---|
ifnot | ⊢ if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnot 142 | . . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑) | |
2 | 1 | iffalsed 4540 | . . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵) |
3 | iftrue 4535 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵) | |
4 | 2, 3 | eqtr4d 2771 | . 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) |
5 | iftrue 4535 | . . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴) | |
6 | iffalse 4538 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴) | |
7 | 5, 6 | eqtr4d 2771 | . 2 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) |
8 | 4, 7 | pm2.61i 182 | 1 ⊢ if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ifcif 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-if 4530 |
This theorem is referenced by: suppsnop 8182 2resupmax 13199 sadadd2lem2 16424 maducoeval2 22541 tmsxpsval2 24447 itg2uba 25672 lgsneg 27253 lgsdilem 27256 sgnneg 34160 bj-xpimasn 36434 itgaddnclem2 37152 ftc1anclem5 37170 |
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