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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 283 | 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 2510 necon2bbid 2982 notzfaus 5360 fressnfv 7159 eluniima 7251 dfac2b 10127 alephval2 10569 adderpqlem 10951 1idpr 11026 leloe 11304 negeq0 11518 addeq0 11641 muleqadd 11862 addltmul 12452 xrleloe 13127 fzrev 13568 mod0 13845 modirr 13911 cjne0 15114 lenegsq 15271 fsumsplit 15691 sumsplit 15718 dvdsabseq 16260 xpsfrnel 17512 isacs2 17601 acsfn 17607 comfeq 17654 sgrp2nmndlem3 18842 resscntz 19238 gexdvds 19493 hauscmplem 23130 hausdiag 23369 utop3cls 23976 affineequivne 26568 eqscut2 27544 ltgov 28115 ax5seglem4 28457 mdsl2i 31842 rspc2daf 31975 cycpmco2 32562 pl1cn 33233 fineqvpow 34394 satefvfmla1 34714 bj-isrvec 36478 topdifinfeq 36534 finxpreclem6 36580 ftc1anclem5 36868 fdc1 36917 relcnveq 37494 relcnveq2 37495 elrelscnveq 37665 elrelscnveq2 37666 lcvexchlem1 38207 lkreqN 38343 glbconxN 38552 islpln5 38709 islvol5 38753 cdlemefrs29bpre0 39570 cdlemg17h 39842 cdlemg33b 39881 tfsconcat0i 42397 tfsconcat0b 42398 oadif1lem 42431 oadif1 42432 brnonrel 42642 frege92 43008 e2ebind 43626 stoweidlem28 45042 |
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