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Mirrors > Home > MPE Home > Th. List > imbi12i | Structured version Visualization version GIF version |
Description: Join two logical equivalences to form equivalence of implications. (Contributed by NM, 1-Aug-1993.) |
Ref | Expression |
---|---|
imbi12i.1 | ⊢ (𝜑 ↔ 𝜓) |
imbi12i.2 | ⊢ (𝜒 ↔ 𝜃) |
Ref | Expression |
---|---|
imbi12i | ⊢ ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbi12i.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
2 | imbi12i.2 | . 2 ⊢ (𝜒 ↔ 𝜃) | |
3 | imbi12 347 | . 2 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))) | |
4 | 1, 2, 3 | mp2 9 | 1 ⊢ ((𝜑 → 𝜒) ↔ (𝜓 → 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: orimdi 930 nanbi 1499 rb-bijust 1752 sbnf2 2355 sb8mo 2596 cbvmowOLD 2599 raleqbii 3339 sbralie 3355 rmo5 3397 cbvrmowOLD 3412 cbvrmo 3426 ss2ab 4057 sbcssg 4524 ssextss 5454 ssrel3 5787 relop 5851 dmcosseq 5973 cotrgOLDOLD 6111 cnvsymOLDOLD 6116 intasym 6117 intirr 6120 codir 6122 qfto 6123 cnvpo 6287 dfpo2 6296 dffun2 6554 dffun2OLD 6555 dff14a 7269 porpss 7717 funcnvuni 7922 poxp 8114 infcllem 9482 ttrclss 9715 cp 9886 aceq2 10114 kmlem12 10156 kmlem15 10159 zfcndpow 10611 grothprim 10829 dfinfre 12195 infrenegsup 12197 xrinfmss2 13290 algcvgblem 16514 isprm2 16619 odulub 18360 oduglb 18362 isirred2 20235 isdomn2 20915 ntreq0 22581 ist0-3 22849 ist1-3 22853 ordthaus 22888 dfconn2 22923 iscusp2 23807 mdsymlem8 31663 mo5f 31729 iuninc 31792 suppss2f 31863 tosglblem 32144 prmidl0 32569 esumpfinvalf 33074 bnj110 33869 bnj92 33873 bnj539 33902 bnj540 33903 axrepprim 34671 axacprim 34676 dffr5 34724 dfso2 34725 elpotr 34753 bj-alcomexcom 35558 itg2addnclem2 36540 isdmn3 36942 sbcimi 36978 inxpss3 37183 trcoss2 37354 moxfr 41430 isdomn3 41946 ifpim123g 42251 elmapintrab 42327 undmrnresiss 42355 cnvssco 42357 snhesn 42537 psshepw 42539 frege77 42691 frege93 42707 frege116 42730 frege118 42732 frege131 42745 frege133 42747 ntrneikb 42845 ismnuprim 43053 onfrALTlem5 43303 onfrALTlem5VD 43646 setis 47743 |
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