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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 136 | . . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑) | |
2 | 1 | iffalsed 4241 | . . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵) |
3 | iftrue 4236 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵) | |
4 | 2, 3 | eqtr4d 2797 | . 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) |
5 | iftrue 4236 | . . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴) | |
6 | iffalse 4239 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴) | |
7 | 5, 6 | eqtr4d 2797 | . 2 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) |
8 | 4, 7 | pm2.61i 176 | 1 ⊢ if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1632 ifcif 4230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-if 4231 |
This theorem is referenced by: suppsnop 7478 2resupmax 12232 sadadd2lem2 15394 maducoeval2 20668 tmsxpsval2 22565 itg2uba 23729 lgsneg 25266 lgsdilem 25269 sgnneg 30932 bj-xpimasn 33266 itgaddnclem2 33800 ftc1anclem5 33820 |
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