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Mirrors > Home > MPE Home > Th. List > orbi2i | Structured version Visualization version GIF version |
Description: Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.) |
Ref | Expression |
---|---|
orbi2i.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
orbi2i | ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orbi2i.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | biimpi 205 | . . 3 ⊢ (𝜑 → 𝜓) |
3 | 2 | orim2i 539 | . 2 ⊢ ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)) |
4 | 1 | biimpri 217 | . . 3 ⊢ (𝜓 → 𝜑) |
5 | 4 | orim2i 539 | . 2 ⊢ ((𝜒 ∨ 𝜓) → (𝜒 ∨ 𝜑)) |
6 | 3, 5 | impbii 198 | 1 ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 |
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 196 df-or 384 |
This theorem is referenced by: orbi1i 541 orbi12i 542 orass 545 or4 549 or42 550 orordir 552 dn1 1000 dfifp6 1012 3orcomb 1041 excxor 1461 nf3 1703 19.44v 1899 19.44 2093 r19.30 3063 sspsstri 3671 unass 3732 undi 3833 undif3 3847 undif3OLD 3848 undif4 3987 ssunpr 4305 sspr 4306 sstp 4307 iinun2 4522 iinuni 4545 qfto 5436 somin1 5448 ordtri2 5675 on0eqel 5762 frxp 7174 wfrlem14 7315 wfrlem15 7316 supgtoreq 8259 wemapsolem 8338 fin1a2lem12 9116 psslinpr 9732 suplem2pr 9754 fimaxre 10847 elnn0 11171 elxnn0 11242 elnn1uz2 11641 elxr 11826 xrinfmss 12012 elfzp1 12261 hashf1lem2 13097 dvdslelem 14869 pythagtrip 15377 tosso 16859 maducoeval2 20265 madugsum 20268 ist0-3 20959 limcdif 23446 ellimc2 23447 limcmpt 23453 limcres 23456 plydivex 23856 taylfval 23917 legtrid 25286 legso 25294 lmicom 25480 nb3graprlem2 25981 numclwwlk3lem 26635 atomli 28625 atoml2i 28626 or3di 28691 disjnf 28766 disjex 28787 disjexc 28788 orngsqr 29135 ind1a 29410 esumcvg 29475 voliune 29619 volfiniune 29620 bnj964 30267 dfso2 30897 socnv 30908 soseq 30995 lineunray 31424 bj-dfbi4 31728 bj-nf3 31767 poimirlem18 32597 poimirlem23 32602 poimirlem27 32606 poimirlem31 32610 itg2addnclem2 32632 tsxo1 33114 tsxo2 33115 tsxo3 33116 tsxo4 33117 tsna1 33121 tsna2 33122 tsna3 33123 ts3an1 33127 ts3an2 33128 ts3an3 33129 ts3or1 33130 ts3or2 33131 ts3or3 33132 ifpim123g 36864 ifpor123g 36872 rp-fakeoranass 36878 frege133d 37076 or3or 37339 undif3VD 38140 wallispilem3 38960 iccpartgt 39965 nnsum4primeseven 40216 nnsum4primesevenALTV 40217 pr1eqbg 40313 nb3grprlem2 40609 av-numclwwlk3lem 41538 lindslinindsimp2 42046 |
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