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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 345 | . 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 927 nanbi 1496 rb-bijust 1749 sbnf2 2352 sb8mo 2593 cbvmowOLD 2596 raleqbii 3336 sbralie 3352 rmo5 3394 cbvrmowOLD 3409 cbvrmo 3423 ss2ab 4055 sbcssg 4522 ssextss 5452 ssrel3 5785 relop 5849 dmcosseq 5971 cotrgOLDOLD 6109 cnvsymOLDOLD 6114 intasym 6115 intirr 6118 codir 6120 qfto 6121 cnvpo 6285 dfpo2 6294 dffun2 6552 dffun2OLD 6553 dff14a 7271 porpss 7719 funcnvuni 7924 poxp 8116 infcllem 9484 ttrclss 9717 cp 9888 aceq2 10116 kmlem12 10158 kmlem15 10161 zfcndpow 10613 grothprim 10831 dfinfre 12199 infrenegsup 12201 xrinfmss2 13294 algcvgblem 16518 isprm2 16623 odulub 18364 oduglb 18366 isirred2 20312 isdomn2 21115 ntreq0 22801 ist0-3 23069 ist1-3 23073 ordthaus 23108 dfconn2 23143 iscusp2 24027 mdsymlem8 31930 mo5f 31996 iuninc 32059 suppss2f 32130 tosglblem 32411 prmidl0 32843 esumpfinvalf 33372 bnj110 34167 bnj92 34171 bnj539 34200 bnj540 34201 axrepprim 34975 axacprim 34980 dffr5 35028 dfso2 35029 elpotr 35057 bj-alcomexcom 35861 itg2addnclem2 36843 isdmn3 37245 sbcimi 37281 inxpss3 37486 trcoss2 37657 moxfr 41732 isdomn3 42248 ifpim123g 42553 elmapintrab 42629 undmrnresiss 42657 cnvssco 42659 snhesn 42839 psshepw 42841 frege77 42993 frege93 43009 frege116 43032 frege118 43034 frege131 43047 frege133 43049 ntrneikb 43147 ismnuprim 43355 onfrALTlem5 43605 onfrALTlem5VD 43948 setis 47830 |
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