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Mirrors > Home > ILE Home > Th. List > 3expd | GIF version |
Description: Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.) |
Ref | Expression |
---|---|
3expd.1 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) |
Ref | Expression |
---|---|
3expd | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3expd.1 | . . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) | |
2 | 1 | com12 30 | . . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜑 → 𝜏)) |
3 | 2 | 3exp 1137 | . 2 ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜑 → 𝜏)))) |
4 | 3 | com4r 85 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 df-3an 921 |
This theorem is referenced by: 3exp2 1156 exp516 1158 3impexp 1366 3impexpbicom 1367 |
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