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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 341 | . 2 ⊢ (𝜑 → ((¬ 𝜃 → 𝜓) ↔ (¬ 𝜃 → 𝜒))) |
3 | df-or 847 | . 2 ⊢ ((𝜃 ∨ 𝜓) ↔ (¬ 𝜃 → 𝜓)) | |
4 | df-or 847 | . 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 205 ∨ wo 846 |
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 847 |
This theorem is referenced by: orbi1d 916 orbi12d 918 eueq2 3707 sbc2or 3787 r19.44zv 4504 rexprgf 4698 rextpg 4704 swopolem 5599 poleloe 6133 elsucg 6433 elsuc2g 6434 xpord2indlem 8133 brdifun 8732 brwdom 9562 isfin1a 10287 elgch 10617 suplem2pr 11048 axlttri 11285 mulcan1g 11867 elznn0 12573 elznn 12574 zindd 12663 rpneg 13006 dfle2 13126 fzm1 13581 fzosplitsni 13743 hashv01gt1 14305 zeo5 16299 bitsf1 16387 lcmval 16529 lcmneg 16540 lcmass 16551 isprm6 16651 infpn2 16846 irredmul 20243 lringuplu 20314 domneq0 20913 znfld 21116 quotval 25805 plydivlem4 25809 plydivex 25810 aalioulem2 25846 aalioulem5 25849 aalioulem6 25850 aaliou 25851 aaliou2 25853 aaliou2b 25854 isinag 28089 axcontlem7 28228 hashecclwwlkn1 29330 elunsn 31750 eliccioo 32097 tlt2 32139 prmidl 32558 mxidlval 32577 sibfof 33339 ballotlemfc0 33491 ballotlemfcc 33492 satfvsucsuc 34356 satf0op 34368 fmlafvel 34376 isfmlasuc 34379 satfv1fvfmla1 34414 seglelin 35088 lineunray 35119 topdifinfeq 36231 wl-ifp4impr 36348 mblfinlem2 36526 pridl 36905 maxidlval 36907 ispridlc 36938 pridlc 36939 dmnnzd 36943 lcfl7N 40372 aomclem8 41803 fzuntgd 42209 orbi1r 43271 iccpartgtl 46094 iccpartleu 46096 lindslinindsimp2lem5 47143 lindslinindsimp2 47144 rrx2pnedifcoorneorr 47403 |
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