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Mirrors > Home > MPE Home > Th. List > bitr3di | Structured version Visualization version GIF version |
Description: A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
Ref | Expression |
---|---|
bitr3di.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
bitr3di.2 | ⊢ (𝜓 ↔ 𝜃) |
Ref | Expression |
---|---|
bitr3di | ⊢ (𝜑 → (𝜒 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bitr3di.2 | . . 3 ⊢ (𝜓 ↔ 𝜃) | |
2 | 1 | bicomi 223 | . 2 ⊢ (𝜃 ↔ 𝜓) |
3 | bitr3di.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | bitr2id 284 | 1 ⊢ (𝜑 → (𝜒 ↔ 𝜃)) |
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: sbco3 2513 necon2bbid 2985 notzfaus 5362 fressnfv 7158 eluniima 7249 dfac2b 10125 alephval2 10567 adderpqlem 10949 1idpr 11024 leloe 11300 negeq0 11514 addeq0 11637 muleqadd 11858 addltmul 12448 xrleloe 13123 fzrev 13564 mod0 13841 modirr 13907 cjne0 15110 lenegsq 15267 fsumsplit 15687 sumsplit 15714 dvdsabseq 16256 xpsfrnel 17508 isacs2 17597 acsfn 17603 comfeq 17650 sgrp2nmndlem3 18806 resscntz 19197 gexdvds 19452 hauscmplem 22910 hausdiag 23149 utop3cls 23756 affineequivne 26332 eqscut2 27307 ltgov 27848 ax5seglem4 28190 mdsl2i 31575 rspc2daf 31708 cycpmco2 32292 pl1cn 32935 fineqvpow 34096 satefvfmla1 34416 bj-isrvec 36175 topdifinfeq 36231 finxpreclem6 36277 ftc1anclem5 36565 fdc1 36614 relcnveq 37191 relcnveq2 37192 elrelscnveq 37362 elrelscnveq2 37363 lcvexchlem1 37904 lkreqN 38040 glbconxN 38249 islpln5 38406 islvol5 38450 cdlemefrs29bpre0 39267 cdlemg17h 39539 cdlemg33b 39578 tfsconcat0i 42095 tfsconcat0b 42096 oadif1lem 42129 oadif1 42130 brnonrel 42340 frege92 42706 e2ebind 43324 stoweidlem28 44744 |
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