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Mirrors > Home > ILE Home > Th. List > impel | GIF version |
Description: An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.) |
Ref | Expression |
---|---|
impel.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
impel.2 | ⊢ (𝜃 → 𝜓) |
Ref | Expression |
---|---|
impel | ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impel.2 | . . 3 ⊢ (𝜃 → 𝜓) | |
2 | impel.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | syl5 32 | . 2 ⊢ (𝜑 → (𝜃 → 𝜒)) |
4 | 3 | imp 124 | 1 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 |
This theorem is referenced by: fiintim 6925 eqinfti 7016 finomni 7135 frecuzrdgrclt 10410 seq3coll 10815 fprodsplitsn 11634 unct 12435 dvcnp2cntop 14034 |
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