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Mirrors > Home > ILE Home > Th. List > animpimp2impd | GIF version |
Description: Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.) |
Ref | Expression |
---|---|
animpimp2impd.1 | ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂))) |
animpimp2impd.2 | ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏)) |
Ref | Expression |
---|---|
animpimp2impd | ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | animpimp2impd.1 | . . . 4 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂))) | |
2 | animpimp2impd.2 | . . . . . 6 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏)) | |
3 | 2 | expr 373 | . . . . 5 ⊢ ((𝜓 ∧ 𝜑) → (𝜃 → (𝜂 → 𝜏))) |
4 | 3 | a2d 26 | . . . 4 ⊢ ((𝜓 ∧ 𝜑) → ((𝜃 → 𝜂) → (𝜃 → 𝜏))) |
5 | 1, 4 | syld 45 | . . 3 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜏))) |
6 | 5 | expcom 115 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) |
7 | 6 | a2d 26 | 1 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem is referenced by: seq3fveq2 10412 seq3shft2 10416 seq3split 10422 seq3id2 10452 |
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