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Mirrors > Home > MPE Home > Th. List > 3an1rs | Structured version Visualization version GIF version |
Description: Swap conjuncts. (Contributed by NM, 16-Dec-2007.) (Proof shortened by Wolf Lammen, 14-Apr-2022.) |
Ref | Expression |
---|---|
3an1rs.1 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
3an1rs | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3an1rs.1 | . . . 4 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | |
2 | 1 | 3exp1 1348 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) |
3 | 2 | com34 91 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 → 𝜏)))) |
4 | 3 | 3imp1 1343 | 1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 |
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 209 df-an 399 df-3an 1085 |
This theorem is referenced by: odf1o2 18700 neiptopnei 21742 cnextcn 22677 nlpineqsn 34691 factwoffsmonot 39105 |
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