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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 144 | . . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑) | |
2 | 1 | iffalsed 4481 | . . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵) |
3 | iftrue 4476 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵) | |
4 | 2, 3 | eqtr4d 2862 | . 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) |
5 | iftrue 4476 | . . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴) | |
6 | iffalse 4479 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴) | |
7 | 5, 6 | eqtr4d 2862 | . 2 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) |
8 | 4, 7 | pm2.61i 184 | 1 ⊢ if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ifcif 4470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-if 4471 |
This theorem is referenced by: suppsnop 7847 2resupmax 12584 sadadd2lem2 15802 maducoeval2 21252 tmsxpsval2 23152 itg2uba 24347 lgsneg 25900 lgsdilem 25903 sgnneg 31802 bj-xpimasn 34271 itgaddnclem2 34955 ftc1anclem5 34975 |
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