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Mirrors > Home > MPE Home > Th. List > orbi2d | Structured version Visualization version GIF version |
Description: Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
bid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
orbi2d | ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bid.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | imbi2d 340 | . 2 ⊢ (𝜑 → ((¬ 𝜃 → 𝜓) ↔ (¬ 𝜃 → 𝜒))) |
3 | df-or 848 | . 2 ⊢ ((𝜃 ∨ 𝜓) ↔ (¬ 𝜃 → 𝜓)) | |
4 | df-or 848 | . 2 ⊢ ((𝜃 ∨ 𝜒) ↔ (¬ 𝜃 → 𝜒)) | |
5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 |
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 207 df-or 848 |
This theorem is referenced by: orbi1d 916 orbi12d 918 eueq2 3718 sbc2or 3799 r19.44zv 4509 elunsn 4687 rexprgf 4699 rextpg 4703 swopolem 5606 poleloe 6153 elsucg 6453 elsuc2g 6454 xpord2indlem 8170 brdifun 8773 brwdom 9604 isfin1a 10329 elgch 10659 suplem2pr 11090 axlttri 11329 mulcan1g 11913 elznn0 12625 elznn 12626 zindd 12716 rpneg 13064 dfle2 13185 fzm1 13643 fzosplitsni 13813 hashv01gt1 14380 zeo5 16389 bitsf1 16479 lcmval 16625 lcmneg 16636 lcmass 16647 isprm6 16747 infpn2 16946 irredmul 20445 lringuplu 20560 domneq0 20724 znfld 21596 quotval 26348 plydivlem4 26352 plydivex 26353 aalioulem2 26389 aalioulem5 26392 aalioulem6 26393 aaliou 26394 aaliou2 26396 aaliou2b 26397 elzs2 28399 elznns 28402 isinag 28860 axcontlem7 28999 hashecclwwlkn1 30105 eliccioo 32897 tlt2 32943 prmidl 33447 mxidlval 33468 rprmdvds 33526 sibfof 34321 ballotlemfc0 34473 ballotlemfcc 34474 satfvsucsuc 35349 satf0op 35361 fmlafvel 35369 isfmlasuc 35372 satfv1fvfmla1 35407 seglelin 36097 lineunray 36128 topdifinfeq 37332 wl-ifp4impr 37449 mblfinlem2 37644 pridl 38023 maxidlval 38025 ispridlc 38056 pridlc 38057 dmnnzd 38061 lcfl7N 41483 aomclem8 43049 fzuntgd 43447 orbi1r 44507 iccpartgtl 47350 iccpartleu 47352 clnbupgrel 47758 dfsclnbgr6 47781 lindslinindsimp2lem5 48307 lindslinindsimp2 48308 rrx2pnedifcoorneorr 48566 |
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