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Mirrors > Home > MPE Home > Th. List > ifeqda | Structured version Visualization version GIF version |
Description: Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.) |
Ref | Expression |
---|---|
ifeqda.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
ifeqda.2 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
ifeqda | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4092 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴) |
3 | ifeqda.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | |
4 | 2, 3 | eqtrd 2656 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐶) |
5 | iffalse 4095 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
6 | 5 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵) |
7 | ifeqda.2 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶) | |
8 | 6, 7 | eqtrd 2656 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐶) |
9 | 4, 8 | pm2.61dan 832 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = 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: somincom 5530 cantnfp1 8578 ccatsymb 13366 swrdccat3blem 13495 repswccat 13532 ccatco 13581 bitsinvp1 15171 xrsdsreval 19791 fvmptnn04if 20654 chfacfscmulgsum 20665 chfacfpmmulgsum 20669 oprpiece1res2 22751 phtpycc 22790 atantayl2 24665 ifeq3da 29365 fprodex01 29571 psgnfzto1stlem 29850 fzto1st1 29852 mdetlap1 29892 madjusmdetlem1 29893 madjusmdetlem2 29894 ccatmulgnn0dir 30619 plymulx 30625 itgexpif 30684 repr0 30689 elmrsubrn 31417 matunitlindflem1 33405 fourierdlem101 40424 hoidmv1lelem2 40806 linc0scn0 42212 m1modmmod 42316 digexp 42401 |
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