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Mirrors > Home > MPE Home > Th. List > ifbothda | Structured version Visualization version GIF version |
Description: A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.) |
Ref | Expression |
---|---|
ifboth.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) |
ifboth.2 | ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) |
ifbothda.3 | ⊢ ((𝜂 ∧ 𝜑) → 𝜓) |
ifbothda.4 | ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒) |
Ref | Expression |
---|---|
ifbothda | ⊢ (𝜂 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbothda.3 | . . 3 ⊢ ((𝜂 ∧ 𝜑) → 𝜓) | |
2 | iftrue 4537 | . . . . . 6 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | 2 | eqcomd 2741 | . . . . 5 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵)) |
4 | ifboth.1 | . . . . 5 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
6 | 5 | adantl 481 | . . 3 ⊢ ((𝜂 ∧ 𝜑) → (𝜓 ↔ 𝜃)) |
7 | 1, 6 | mpbid 232 | . 2 ⊢ ((𝜂 ∧ 𝜑) → 𝜃) |
8 | ifbothda.4 | . . 3 ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒) | |
9 | iffalse 4540 | . . . . . 6 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
10 | 9 | eqcomd 2741 | . . . . 5 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵)) |
11 | ifboth.2 | . . . . 5 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜃)) |
13 | 12 | adantl 481 | . . 3 ⊢ ((𝜂 ∧ ¬ 𝜑) → (𝜒 ↔ 𝜃)) |
14 | 8, 13 | mpbid 232 | . 2 ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜃) |
15 | 7, 14 | pm2.61dan 813 | 1 ⊢ (𝜂 → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ifcif 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-if 4532 |
This theorem is referenced by: ifboth 4570 resixpfo 8975 boxriin 8979 boxcutc 8980 suppr 9509 infpr 9541 cantnflem1 9727 ttukeylem5 10551 ttukeylem6 10552 bitsinv1lem 16475 bitsinv1 16476 smumullem 16526 hashgcdeq 16823 ramcl2lem 17043 acsfn 17704 tsrlemax 18644 odlem1 19568 gexlem1 19612 cyggex2 19930 dprdfeq0 20057 xrsdsreclb 21449 mplmon2 22103 evlslem1 22124 coe1tmmul2 22295 coe1tmmul 22296 ptcld 23637 xkopt 23679 stdbdxmet 24544 xrsxmet 24845 iccpnfcnv 24989 iccpnfhmeo 24990 xrhmeo 24991 dvcobr 25998 dvcobrOLD 25999 mdegle0 26131 dvradcnv 26479 psercnlem1 26484 psercn 26485 logtayl 26717 efrlim 27027 efrlimOLD 27028 lgamgulmlem5 27091 musum 27249 dchrmullid 27311 dchrsum2 27327 sumdchr2 27329 dchrisum0flblem1 27567 dchrisum0flblem2 27568 rplogsum 27586 pntlemj 27662 eupth2lem1 30247 eulerpathpr 30269 ifeqeqx 32563 elrgspnlem2 33233 elrgspnlem3 33234 xrge0iifcnv 33894 xrge0iifhom 33898 esumpinfval 34054 dstfrvunirn 34456 sgn3da 34523 plymulx0 34541 signswn0 34554 signswch 34555 lpadmax 34676 lpadright 34678 fnejoin2 36352 poimirlem16 37623 poimirlem17 37624 poimirlem19 37626 poimirlem20 37627 poimirlem24 37631 cnambfre 37655 itg2addnclem 37658 itg2addnclem3 37660 itg2addnc 37661 itg2gt0cn 37662 ftc1anclem7 37686 ftc1anclem8 37687 ftc1anc 37688 sticksstones10 42137 sticksstones12a 42139 aks6d1c6lem3 42154 metakunt1 42187 metakunt2 42188 metakunt25 42211 kelac1 43052 |
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