Theorem 3simpb 998

Description: Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)

Assertion

Ref	Expression
3simpb	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))

Proof of Theorem 3simpb

Step	Hyp	Ref	Expression
1		3ancomb 989	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓))
2		3simpa 997	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → (𝜑 ∧ 𝜒))
3	1, 2	sylbi 121	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 983

This theorem is referenced by:  3adant2  1019  3adantl2  1157  3adantr2  1160  enq0tr  7560  ixxssixx  10037  rebtwn2zlemshrink  10409  zsumdc  11745  muldvds1  12177  dvds2add  12186  dvds2sub  12187  dvdstr  12189  pw2dvdslemn  12537  ctinf  12851  mndissubm  13357  gsumfzconst  13727