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| Mirrors > Home > MPE Home > Th. List > ad4ant134 | Structured version Visualization version GIF version | ||
| Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) | 
| Ref | Expression | 
|---|---|
| ad4ant3.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | 
| Ref | Expression | 
|---|---|
| ad4ant134 | ⊢ ((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ad4ant3.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
| 2 | 1 | 3expa 1118 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) | 
| 3 | 2 | adantllr 719 | 1 ⊢ ((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 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: ad5ant245 1362 ad5ant134 1368 ad5ant135 1369 ad5ant145 1370 ralxfrd2 5411 gruwun 10854 lemul12b 12125 initoeu1 18057 termoeu1 18064 quscrng 21294 metss 24522 wlkswwlksf1o 29900 climxlim2lem 45865 smflimlem4 46794 isubgr3stgrlem8 47945 | 
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