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Mirrors > Home > MPE Home > Th. List > orbi1d | Structured version Visualization version GIF version |
Description: Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
bid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
orbi1d | ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bid.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | orbi2d 914 | . 2 ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒))) |
3 | orcom 868 | . 2 ⊢ ((𝜓 ∨ 𝜃) ↔ (𝜃 ∨ 𝜓)) | |
4 | orcom 868 | . 2 ⊢ ((𝜒 ∨ 𝜃) ↔ (𝜃 ∨ 𝜒)) | |
5 | 2, 3, 4 | 3bitr4g 313 | 1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 |
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 df-or 846 |
This theorem is referenced by: orbi1 916 orbi12d 917 eueq2 3697 uneq1 4147 r19.45zv 4491 rexprgf 4685 rextpg 4691 swopolem 5586 ordsseleq 6377 ordtri3 6384 frxp2 8107 xpord2pred 8108 xpord2indlem 8110 frxp3 8114 xpord3pred 8115 infltoreq 9474 cantnflem1 9661 axgroth2 10797 axgroth3 10803 lelttric 11298 ltxr 13072 xmulneg1 13225 fzpr 13533 elfzp12 13557 caubnd 15282 lcmval 16506 lcmass 16528 isprm6 16628 vdwlem10 16900 irredmul 20188 lringuplu 20259 domneq0 20842 znfld 21042 opsrval 21522 logreclem 26187 perfectlem2 26653 legov3 27660 lnhl 27677 colperpex 27795 lmif 27847 islmib 27849 friendshipgt3 29462 h1datom 30644 xrlelttric 31782 tlt3 31957 prmidl 32352 ismxidl 32372 esumpcvgval 32842 sibfof 33105 satfvsuc 34118 satfv1 34120 satfvsucsuc 34122 satf0suc 34133 sat1el2xp 34136 fmlasuc0 34141 fmlafvel 34142 satfv1fvfmla1 34180 segcon2 34842 wl-ifpimpr 36086 poimirlem25 36252 cnambfre 36275 pridl 36645 ismaxidl 36648 ispridlc 36678 pridlc 36679 dmnnzd 36683 disjecxrncnvep 37000 4atlem3a 38208 pmapjoin 38463 lcfl3 40105 lcfl4N 40106 sticksstones22 40724 ordsssucb 41793 sbcoreleleqVD 43328 fourierdlem80 44612 euoreqb 45526 el1fzopredsuc 45742 perfectALTVlem2 46099 nnsum3primesle9 46171 lindslinindsimp2 46729 |
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