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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 219, sylib 220, sylbir 237, sylibr 236; four inferences inferring an implication from two biconditionals: sylbb 221, sylbbr 238, sylbb1 239, sylbb2 240; four inferences inferring a biconditional from two biconditionals: bitri 277, bitr2i 278, bitr3i 279, bitr4i 280 (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 234, bitrd 281, syl5bb 285, syl6bb 289 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 230 | . 2 ⊢ (𝜒 → 𝜓) |
3 | sylbbr.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
4 | 2, 3 | sylibr 236 | 1 ⊢ (𝜒 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 |
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 209 |
This theorem is referenced by: bitri 277 euelss 4278 dfnfc2 4846 ndmima 5952 axcclem 9865 cshw1 14169 fsumcom2 15114 fprodcom2 15323 pmtr3ncomlem1 18584 mdetunilem7 21210 cmpcov2 21981 hausflf2 22589 umgredg 26909 vtxdginducedm1 27311 2pthfrgrrn 28045 eqdif 30267 cusgredgex2 32376 conway 33271 f1omptsnlem 34633 igenval2 35376 mpobi123f 35472 brtrclfv2 40162 clsk1indlem3 40483 or2expropbilem1 43357 |
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