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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 224 | . 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 206 |
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 207 |
This theorem is referenced by: sbco3 2516 necon2bbid 2982 notzfaus 5369 fressnfv 7180 eluniima 7270 dfac2b 10169 alephval2 10610 adderpqlem 10992 1idpr 11067 leloe 11345 negeq0 11561 addeq0 11684 muleqadd 11905 addltmul 12500 xrleloe 13183 fzrev 13624 mod0 13913 modirr 13980 cjne0 15199 lenegsq 15356 fsumsplit 15774 sumsplit 15801 dvdsabseq 16347 xpsfrnel 17609 isacs2 17698 acsfn 17704 comfeq 17751 sgrp2nmndlem3 18951 resscntz 19364 gexdvds 19617 hauscmplem 23430 hausdiag 23669 utop3cls 24276 affineequivne 26885 eqscut2 27866 ltgov 28620 ax5seglem4 28962 mdsl2i 32351 rspc2daf 32495 cycpmco2 33136 pl1cn 33916 fineqvpow 35089 satefvfmla1 35410 bj-isrvec 37277 topdifinfeq 37333 finxpreclem6 37379 wl-sb8ft 37531 ftc1anclem5 37684 fdc1 37733 relcnveq 38304 relcnveq2 38305 elrelscnveq 38474 elrelscnveq2 38475 lcvexchlem1 39016 lkreqN 39152 glbconxN 39361 islpln5 39518 islvol5 39562 cdlemefrs29bpre0 40379 cdlemg17h 40651 cdlemg33b 40690 tfsconcat0i 43335 tfsconcat0b 43336 oadif1lem 43369 oadif1 43370 brnonrel 43579 frege92 43945 e2ebind 44561 stoweidlem28 45984 clnbupgrel 47759 |
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