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Mirrors > Home > ILE Home > Th. List > 3an1rs | 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 115 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜃 → 𝜏)) |
3 | 2 | 3exp 1202 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) |
4 | 3 | com34 83 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 → 𝜏)))) |
5 | 4 | 3imp 1193 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → (𝜒 → 𝜏)) |
6 | 5 | imp 124 | 1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 df-3an 980 |
This theorem is referenced by: (None) |
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