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Mirrors > Home > MPE Home > Th. List > elimhyp | Structured version Visualization version GIF version |
Description: Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4089. (Contributed by NM, 15-May-1999.) |
Ref | Expression |
---|---|
elimhyp.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓)) |
elimhyp.2 | ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓)) |
elimhyp.3 | ⊢ 𝜒 |
Ref | Expression |
---|---|
elimhyp | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4042 | . . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | eqcomd 2616 | . . . 4 ⊢ (𝜑 → 𝐴 = if(𝜑, 𝐴, 𝐵)) |
3 | elimhyp.1 | . . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → (𝜑 ↔ 𝜓)) |
5 | 4 | ibi 255 | . 2 ⊢ (𝜑 → 𝜓) |
6 | elimhyp.3 | . . 3 ⊢ 𝜒 | |
7 | iffalse 4045 | . . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
8 | 7 | eqcomd 2616 | . . . 4 ⊢ (¬ 𝜑 → 𝐵 = if(𝜑, 𝐴, 𝐵)) |
9 | elimhyp.2 | . . . 4 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (¬ 𝜑 → (𝜒 ↔ 𝜓)) |
11 | 6, 10 | mpbii 222 | . 2 ⊢ (¬ 𝜑 → 𝜓) |
12 | 5, 11 | pm2.61i 175 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ifcif 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-if 4037 |
This theorem is referenced by: elimel 4100 elimf 5957 oeoa 7564 oeoe 7566 limensuc 8022 axcc4dom 9146 elimne0 9909 elimgt0 10738 elimge0 10739 2ndcdisj 21069 siilem2 27091 normlem7tALT 27360 hhsssh 27510 shintcl 27573 chintcl 27575 spanun 27788 elunop2 28256 lnophm 28262 nmbdfnlb 28293 hmopidmch 28396 hmopidmpj 28397 chirred 28638 limsucncmp 31615 elimhyps 33265 elimhyps2 33268 |
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