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Mirrors > Home > ILE Home > Th. List > jaodan | GIF version |
Description: Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
jaodan.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
jaodan.2 | ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
Ref | Expression |
---|---|
jaodan | ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃)) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jaodan.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 114 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | jaodan.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) | |
4 | 3 | ex 114 | . . 3 ⊢ (𝜑 → (𝜃 → 𝜒)) |
5 | 2, 4 | jaod 706 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → 𝜒)) |
6 | 5 | imp 123 | 1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃)) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: mpjaodan 787 ordi 805 andi 807 dcor 919 ccase 948 mpjao3dan 1285 relop 4684 poltletr 4934 tfrlemisucaccv 6215 tfr1onlemsucaccv 6231 tfrcllemsucaccv 6244 phplem3 6741 ssfilem 6762 diffitest 6774 reapmul1 8350 apsqgt0 8356 recexaplem2 8406 nnnn0addcl 9000 un0addcl 9003 un0mulcl 9004 elz2 9115 xrltso 9575 xaddnemnf 9633 xaddnepnf 9634 fzsplit2 9823 fzsuc2 9852 elfzp12 9872 expp1 10293 expnegap0 10294 expcllem 10297 mulexpzap 10326 expaddzap 10330 expmulzap 10332 bcpasc 10505 xrltmaxsup 11019 xrmaxaddlem 11022 summodc 11145 fsumsplit 11169 ef0lem 11355 odd2np1 11559 dvdslcm 11739 lcmeq0 11741 lcmcl 11742 lcmneg 11744 lcmgcd 11748 rpexp1i 11821 |
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