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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 220, sylib 221, sylbir 238, sylibr 237; four inferences inferring an implication from two biconditionals: sylbb 222, sylbbr 239, sylbb1 240, sylbb2 241; four inferences inferring a biconditional from two biconditionals: bitri 278, bitr2i 279, bitr3i 280, bitr4i 281 (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 235, bitrd 282, syl5bb 286, syl6bb 290 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 231 | . 2 ⊢ (𝜒 → 𝜓) |
3 | sylbbr.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
4 | 2, 3 | sylibr 237 | 1 ⊢ (𝜒 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 |
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 210 |
This theorem is referenced by: bitri 278 euelss 4242 dfnfc2 4822 ndmima 5933 axcclem 9868 cshw1 14175 fsumcom2 15121 fprodcom2 15330 pmtr3ncomlem1 18593 mdetunilem7 21223 cmpcov2 21995 hausflf2 22603 umgredg 26931 vtxdginducedm1 27333 2pthfrgrrn 28067 eqdif 30290 cusgredgex2 32482 conway 33377 f1omptsnlem 34753 igenval2 35504 mpobi123f 35600 brtrclfv2 40428 clsk1indlem3 40746 or2expropbilem1 43624 |
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