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Mirrors > Home > ILE Home > Th. List > 3simpc | GIF version |
Description: Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Ref | Expression |
---|---|
3simpc | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anrot 890 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑)) | |
2 | 3simpa 901 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) → (𝜓 ∧ 𝜒)) | |
3 | 1, 2 | sylbi 114 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 885 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 df-3an 887 |
This theorem is referenced by: simp3 906 3adant1 922 3adantl1 1060 3adantr1 1063 eupickb 1981 find 4322 divcanap2 7659 diveqap0 7661 divrecap 7667 divcanap3 7675 eliooord 8797 fzrev3 8949 sqdivap 9318 |
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