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Mirrors > Home > MPE Home > Th. List > ifboth | 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, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.) |
Ref | Expression |
---|---|
ifboth.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) |
ifboth.2 | ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
ifboth | ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifboth.1 | . 2 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) | |
2 | ifboth.2 | . 2 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) | |
3 | simpll 766 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜓) | |
4 | simplr 768 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ ¬ 𝜑) → 𝜒) | |
5 | 1, 2, 3, 4 | ifbothda 4562 | 1 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ifcif 4524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-if 4525 |
This theorem is referenced by: ifcl 4569 keephyp 4595 soltmin 6136 xrmaxlt 13184 xrltmin 13185 xrmaxle 13186 xrlemin 13187 ifle 13200 expmulnbnd 14221 limsupgre 15449 isumless 15815 cvgrat 15853 rpnnen2lem4 16185 ruclem2 16200 sadcaddlem 16423 sadadd3 16427 pcmptdvds 16854 prmreclem5 16880 prmreclem6 16881 pnfnei 23111 mnfnei 23112 xkopt 23546 xmetrtri2 24249 stdbdxmet 24411 stdbdmet 24412 stdbdmopn 24414 xrsxmet 24712 icccmplem2 24726 metdscn 24759 metnrmlem1a 24761 ivthlem2 25368 ovolicc2lem5 25437 ioombl1lem1 25474 ioombl1lem4 25477 ismbfd 25555 mbfi1fseqlem4 25635 mbfi1fseqlem5 25636 itg2const 25657 itg2const2 25658 itg2monolem3 25669 itg2gt0 25677 itg2cnlem1 25678 itg2cnlem2 25679 iblss 25721 itgless 25733 ibladdlem 25736 iblabsr 25746 iblmulc2 25747 bddiblnc 25758 dvferm1lem 25903 dvferm2lem 25905 dvlip2 25915 dgradd2 26190 plydiveu 26220 chtppilim 27395 dchrvmasumiflem1 27421 ostth3 27558 1smat1 33341 poimirlem24 37052 mblfinlem2 37066 itg2addnclem 37079 itg2addnc 37082 itg2gt0cn 37083 ibladdnclem 37084 iblmulc2nc 37093 ftc1anclem5 37105 ftc1anclem8 37108 ftc1anc 37109 |
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