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Mirrors > Home > MPE Home > Th. List > sylbbr | Structured version Visualization version GIF version |
Description: A mixed syllogism
inference from two biconditionals.
Note on the various syllogism-like statements in set.mm. The hypothetical syllogism syl 17 infers an implication from two implications (and there are 3syl 18 and 4syl 19 for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: sylbi 216, sylib 217, sylbir 234, sylibr 233; four inferences inferring an implication from two biconditionals: sylbb 218, sylbbr 235, sylbb1 236, sylbb2 237; four inferences inferring a biconditional from two biconditionals: bitri 275, bitr2i 276, bitr3i 277, bitr4i 278 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 47, syl5 34, syl6 35, mpbid 231, bitrd 279, bitrid 283, bitrdi 287 and variants. (Contributed by BJ, 21-Apr-2019.) |
Ref | Expression |
---|---|
sylbbr.1 | ⊢ (𝜑 ↔ 𝜓) |
sylbbr.2 | ⊢ (𝜓 ↔ 𝜒) |
Ref | Expression |
---|---|
sylbbr | ⊢ (𝜒 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylbbr.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
2 | 1 | biimpri 227 | . 2 ⊢ (𝜒 → 𝜓) |
3 | sylbbr.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
4 | 2, 3 | sylibr 233 | 1 ⊢ (𝜒 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 |
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 206 |
This theorem is referenced by: bitri 275 euelss 4322 dfnfc2 4934 ndmima 6103 unfi 9172 axcclem 10452 cshw1 14772 fsumcom2 15720 fprodcom2 15928 pmtr3ncomlem1 19341 mdetunilem7 22120 cmpcov2 22894 hausflf2 23502 conway 27301 umgredg 28429 vtxdginducedm1 28831 2pthfrgrrn 29566 eqdif 31788 cusgredgex2 34144 f1omptsnlem 36265 igenval2 36982 mpobi123f 37078 brtrclfv2 42526 clsk1indlem3 42842 or2expropbilem1 45790 mo0sn 47548 |
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