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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 4584. (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 4531 | . . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
| 2 | 1 | eqcomd 2743 | . . . 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 267 | . 2 ⊢ (𝜑 → 𝜓) |
| 6 | elimhyp.3 | . . 3 ⊢ 𝜒 | |
| 7 | iffalse 4534 | . . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
| 8 | 7 | eqcomd 2743 | . . . 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 233 | . 2 ⊢ (¬ 𝜑 → 𝜓) |
| 12 | 5, 11 | pm2.61i 182 | 1 ⊢ 𝜓 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ifcif 4525 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4526 |
| This theorem is referenced by: elimel 4595 elimf 6735 oeoa 8635 oeoe 8637 limensuc 9194 axcc4dom 10481 elimne0 11251 elimgt0 12105 elimge0 12106 2ndcdisj 23464 siilem2 30871 normlem7tALT 31138 hhsssh 31288 shintcl 31349 chintcl 31351 spanun 31564 elunop2 32032 lnophm 32038 nmbdfnlb 32069 hmopidmch 32172 hmopidmpj 32173 chirred 32414 limsucncmp 36447 elimhyps 38962 elimhyps2 38965 |
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