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Mirrors > Home > MPE Home > Th. List > ifeq1da | Structured version Visualization version GIF version |
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ifeq1da.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ifeq1da | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1da.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) | |
2 | 1 | ifeq1d 4510 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
3 | iffalse 4500 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶) | |
4 | iffalse 4500 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶) | |
5 | 3, 4 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
6 | 5 | adantl 483 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
7 | 2, 6 | pm2.61dan 812 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ifcif 4491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-un 3920 df-if 4492 |
This theorem is referenced by: ifeq12da 4524 cantnflem1d 9631 cantnflem1 9632 dfac12lem1 10086 xrmaxeq 13105 xrmineq 13106 rexmul 13197 max0add 15202 sumeq2ii 15585 fsumser 15622 ramcl 16908 dmdprdsplitlem 19823 coe1pwmul 21666 scmatscmiddistr 21873 mulmarep1gsum1 21938 maducoeval2 22005 madugsum 22008 madurid 22009 ptcld 22980 ibllem 25145 itgvallem3 25166 iblposlem 25172 iblss2 25186 iblmulc2 25211 cnplimc 25267 limcco 25273 dvexp3 25358 dchrinvcl 26617 lgsval2lem 26671 lgsval4lem 26672 lgsneg 26685 lgsmod 26687 lgsdilem2 26697 rpvmasum2 26876 mrsubrn 34147 ftc1anclem6 36185 ftc1anclem8 36187 fsuppind 40794 |
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