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Mirrors > Home > ILE Home > Th. List > 3simpb | GIF version |
Description: Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) |
Ref | Expression |
---|---|
3simpb | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3ancomb 981 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓)) | |
2 | 3simpa 989 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → (𝜑 ∧ 𝜒)) | |
3 | 1, 2 | sylbi 120 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 |
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 depends on definitions: df-bi 116 df-3an 975 |
This theorem is referenced by: 3adant2 1011 3adantl2 1149 3adantr2 1152 enq0tr 7383 ixxssixx 9846 rebtwn2zlemshrink 10197 zsumdc 11334 muldvds1 11765 dvds2add 11774 dvds2sub 11775 dvdstr 11777 pw2dvdslemn 12106 ctinf 12372 mndissubm 12683 |
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