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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 4587. (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 4535 | . . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | eqcomd 2739 | . . . 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 4538 | . . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
8 | 7 | eqcomd 2739 | . . . 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 232 | . 2 ⊢ (¬ 𝜑 → 𝜓) |
12 | 5, 11 | pm2.61i 182 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ifcif 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-if 4530 |
This theorem is referenced by: elimel 4598 elimf 6717 oeoa 8597 oeoe 8599 limensuc 9154 axcc4dom 10436 elimne0 11204 elimgt0 12052 elimge0 12053 2ndcdisj 22960 siilem2 30105 normlem7tALT 30372 hhsssh 30522 shintcl 30583 chintcl 30585 spanun 30798 elunop2 31266 lnophm 31272 nmbdfnlb 31303 hmopidmch 31406 hmopidmpj 31407 chirred 31648 limsucncmp 35331 elimhyps 37831 elimhyps2 37834 |
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