![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 764 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜓) | |
4 | simplr 766 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ ¬ 𝜑) → 𝜒) | |
5 | 1, 2, 3, 4 | ifbothda 4566 | 1 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ifcif 4528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-if 4529 |
This theorem is referenced by: ifcl 4573 keephyp 4599 soltmin 6137 xrmaxlt 13167 xrltmin 13168 xrmaxle 13169 xrlemin 13170 ifle 13183 expmulnbnd 14205 limsupgre 15432 isumless 15798 cvgrat 15836 rpnnen2lem4 16167 ruclem2 16182 sadcaddlem 16405 sadadd3 16409 pcmptdvds 16834 prmreclem5 16860 prmreclem6 16861 pnfnei 23044 mnfnei 23045 xkopt 23479 xmetrtri2 24182 stdbdxmet 24344 stdbdmet 24345 stdbdmopn 24347 xrsxmet 24645 icccmplem2 24659 metdscn 24692 metnrmlem1a 24694 ivthlem2 25301 ovolicc2lem5 25370 ioombl1lem1 25407 ioombl1lem4 25410 ismbfd 25488 mbfi1fseqlem4 25568 mbfi1fseqlem5 25569 itg2const 25590 itg2const2 25591 itg2monolem3 25602 itg2gt0 25610 itg2cnlem1 25611 itg2cnlem2 25612 iblss 25654 itgless 25666 ibladdlem 25669 iblabsr 25679 iblmulc2 25680 bddiblnc 25691 dvferm1lem 25836 dvferm2lem 25838 dvlip2 25848 dgradd2 26121 plydiveu 26150 chtppilim 27321 dchrvmasumiflem1 27347 ostth3 27484 1smat1 33248 poimirlem24 36976 mblfinlem2 36990 itg2addnclem 37003 itg2addnc 37006 itg2gt0cn 37007 ibladdnclem 37008 iblmulc2nc 37017 ftc1anclem5 37029 ftc1anclem8 37032 ftc1anc 37033 |
Copyright terms: Public domain | W3C validator |