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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 3699 uneq1 4149 r19.45zv 4493 rexprgf 4687 rextpg 4693 swopolem 5588 ordsseleq 6379 ordtri3 6386 frxp2 8109 xpord2pred 8110 xpord2indlem 8112 frxp3 8116 xpord3pred 8117 infltoreq 9476 cantnflem1 9663 axgroth2 10799 axgroth3 10805 lelttric 11300 ltxr 13074 xmulneg1 13227 fzpr 13535 elfzp12 13559 caubnd 15284 lcmval 16508 lcmass 16530 isprm6 16630 vdwlem10 16902 irredmul 20190 lringuplu 20261 domneq0 20844 znfld 21044 opsrval 21524 logreclem 26189 perfectlem2 26655 legov3 27709 lnhl 27726 colperpex 27844 lmif 27896 islmib 27898 friendshipgt3 29511 h1datom 30693 xrlelttric 31831 tlt3 32006 prmidl 32401 ismxidl 32421 esumpcvgval 32891 sibfof 33154 satfvsuc 34167 satfv1 34169 satfvsucsuc 34171 satf0suc 34182 sat1el2xp 34185 fmlasuc0 34190 fmlafvel 34191 satfv1fvfmla1 34229 segcon2 34891 wl-ifpimpr 36135 poimirlem25 36301 cnambfre 36324 pridl 36694 ismaxidl 36697 ispridlc 36727 pridlc 36728 dmnnzd 36732 disjecxrncnvep 37049 4atlem3a 38257 pmapjoin 38512 lcfl3 40154 lcfl4N 40155 sticksstones22 40773 ordsssucb 41842 sbcoreleleqVD 43377 fourierdlem80 44661 euoreqb 45575 el1fzopredsuc 45791 perfectALTVlem2 46148 nnsum3primesle9 46220 lindslinindsimp2 46778 |
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