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Mirrors > Home > ILE Home > Th. List > anabsan | GIF version |
Description: Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.) |
Ref | Expression |
---|---|
anabsan.1 | ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
anabsan | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.24 390 | . 2 ⊢ (𝜑 ↔ (𝜑 ∧ 𝜑)) | |
2 | anabsan.1 | . 2 ⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒) | |
3 | 1, 2 | sylanb 280 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: anabss1 548 anabss5 550 anandis 564 iddvds 11351 1dvds 11352 |
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