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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 4097 | . . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵) |
3 | iftrue 4092 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵) | |
4 | 2, 3 | eqtr4d 2659 | . 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) |
5 | iftrue 4092 | . . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴) | |
6 | iffalse 4095 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴) | |
7 | 5, 6 | eqtr4d 2659 | . 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 1483 ifcif 4086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-if 4087 |
This theorem is referenced by: suppsnop 7309 2resupmax 12019 sadadd2lem2 15172 maducoeval2 20446 tmsxpsval2 22344 itg2uba 23510 lgsneg 25046 lgsdilem 25049 sgnneg 30602 bj-xpimasn 32942 itgaddnclem2 33469 ftc1anclem5 33489 |
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