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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 767 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜑) → 𝜓) | |
4 | simplr 769 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ ¬ 𝜑) → 𝜒) | |
5 | 1, 2, 3, 4 | ifbothda 4568 | 1 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ifcif 4530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-if 4531 |
This theorem is referenced by: ifcl 4575 keephyp 4601 soltmin 6158 xrmaxlt 13219 xrltmin 13220 xrmaxle 13221 xrlemin 13222 ifle 13235 expmulnbnd 14270 limsupgre 15513 isumless 15877 cvgrat 15915 rpnnen2lem4 16249 ruclem2 16264 sadcaddlem 16490 sadadd3 16494 pcmptdvds 16927 prmreclem5 16953 prmreclem6 16954 pnfnei 23243 mnfnei 23244 xkopt 23678 xmetrtri2 24381 stdbdxmet 24543 stdbdmet 24544 stdbdmopn 24546 xrsxmet 24844 icccmplem2 24858 metdscn 24891 metnrmlem1a 24893 ivthlem2 25500 ovolicc2lem5 25569 ioombl1lem1 25606 ioombl1lem4 25609 ismbfd 25687 mbfi1fseqlem4 25767 mbfi1fseqlem5 25768 itg2const 25789 itg2const2 25790 itg2monolem3 25801 itg2gt0 25809 itg2cnlem1 25810 itg2cnlem2 25811 iblss 25854 itgless 25866 ibladdlem 25869 iblabsr 25879 iblmulc2 25880 bddiblnc 25891 dvferm1lem 26036 dvferm2lem 26038 dvlip2 26048 dgradd2 26322 plydiveu 26354 chtppilim 27533 dchrvmasumiflem1 27559 ostth3 27696 1smat1 33764 poimirlem24 37630 mblfinlem2 37644 itg2addnclem 37657 itg2addnc 37660 itg2gt0cn 37661 ibladdnclem 37662 iblmulc2nc 37671 ftc1anclem5 37683 ftc1anclem8 37686 ftc1anc 37687 |
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