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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.) |
Ref | Expression |
---|---|
3an1rs.1 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
3an1rs | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3an1rs.1 | . . . . . 6 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | |
2 | 1 | ex 449 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏)) |
3 | 2 | 3exp 1256 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) |
4 | 3 | com34 89 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 → 𝜏)))) |
5 | 4 | 3imp 1249 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → (𝜒 → 𝜏)) |
6 | 5 | imp 444 | 1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 196 df-an 385 df-3an 1033 |
This theorem is referenced by: odf1o2 17811 neiptopnei 20746 cnextcn 21681 |
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