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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 4312 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | adantl 475 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴) |
3 | ifeqda.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | |
4 | 2, 3 | eqtrd 2813 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐶) |
5 | iffalse 4315 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵) | |
6 | 5 | adantl 475 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵) |
7 | ifeqda.2 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶) | |
8 | 6, 7 | eqtrd 2813 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐶) |
9 | 4, 8 | pm2.61dan 803 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ifcif 4306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-if 4307 |
This theorem is referenced by: somincom 5785 cantnfp1 8875 ccatsymb 13672 swrdccat3blem 13871 repswccat 13932 ccatco 13986 bitsinvp1 15577 xrsdsreval 20187 fvmptnn04if 21061 chfacfscmulgsum 21072 chfacfpmmulgsum 21076 oprpiece1res2 23159 phtpycc 23198 atantayl2 25116 ifeq3da 29928 fprodex01 30135 psgnfzto1stlem 30448 fzto1st1 30450 mdetlap1 30490 madjusmdetlem1 30491 madjusmdetlem2 30492 ccatmulgnn0dir 31219 plymulx 31225 itgexpif 31286 repr0 31291 elmrsubrn 32016 matunitlindflem1 34026 fourierdlem101 41344 hoidmv1lelem2 41726 dfafv2 42166 linc0scn0 43220 m1modmmod 43324 digexp 43409 |
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