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| Mirrors > Home > MPE Home > Th. List > 3adant2r | Structured version Visualization version GIF version | ||
| Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.) | 
| Ref | Expression | 
|---|---|
| ad4ant3.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | 
| Ref | Expression | 
|---|---|
| 3adant2r | ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . 2 ⊢ ((𝜓 ∧ 𝜏) → 𝜓) | |
| 2 | ad4ant3.1 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
| 3 | 1, 2 | syl3an2 1164 | 1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 | 
| 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-an 396 df-3an 1088 | 
| This theorem is referenced by: ltdiv23 12160 lediv23 12161 divalglem8 16438 isdrngd 20766 isdrngdOLD 20768 deg1tm 26159 ax5seglem1 28944 ax5seglem2 28945 nvaddsub4 30677 nmoub2i 30794 cdleme21at 40331 cdleme42f 40483 trlcoabs2N 40725 tendoplcl2 40781 tendopltp 40783 cdlemk2 40835 cdlemk8 40841 cdlemk9 40842 cdlemk9bN 40843 cdleml8 40986 dihglblem3N 41298 dihglblem3aN 41299 fourierdlem42 46169 lincscm 48352 itsclc0yqsol 48690 | 
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