Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4478 | . . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵) |
3 | iftrue 4473 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵) | |
4 | 2, 3 | eqtr4d 2859 | . 2 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)) |
5 | iftrue 4473 | . . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴) | |
6 | iffalse 4476 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴) | |
7 | 5, 6 | eqtr4d 2859 | . 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 1537 ifcif 4467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-if 4468 |
This theorem is referenced by: suppsnop 7844 2resupmax 12582 sadadd2lem2 15799 maducoeval2 21249 tmsxpsval2 23149 itg2uba 24344 lgsneg 25897 lgsdilem 25900 sgnneg 31798 bj-xpimasn 34270 itgaddnclem2 34966 ftc1anclem5 34986 |
Copyright terms: Public domain | W3C validator |