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Mirrors > Home > MPE Home > Th. List > 3bitr2rd | Structured version Visualization version GIF version |
Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) |
Ref | Expression |
---|---|
3bitr2d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
3bitr2d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜒)) |
3bitr2d.3 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Ref | Expression |
---|---|
3bitr2rd | ⊢ (𝜑 → (𝜏 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitr2d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 3bitr2d.2 | . . 3 ⊢ (𝜑 → (𝜃 ↔ 𝜒)) | |
3 | 1, 2 | bitr4d 270 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
4 | 3bitr2d.3 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
5 | 3, 4 | bitr2d 268 | 1 ⊢ (𝜑 → (𝜏 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 |
This theorem is referenced by: fnsuppres 7209 addsubeq4 10175 muleqadd 10550 mulle0b 10773 adddivflid 12481 om2uzlti 12611 summodnegmod 14850 qnumdenbi 15290 dprdf11 18245 lvecvscan2 18933 mdetunilem9 20245 elfilss 21490 itg2seq 23315 itg2cnlem2 23335 chpchtsum 24744 bposlem7 24815 lgsdilem 24849 lgsne0 24860 colhp 25462 axcontlem7 25650 pjnorm2 27970 cdj3lem1 28677 zrhchr 29348 dochfln0 35784 mapdindp 35978 |
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