Theorem 3simpb 1021

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 1012	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓))
2		3simpa 1020	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → (𝜑 ∧ 𝜒))
3	1, 2	sylbi 121	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004

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 1006

This theorem is referenced by:  3adant2  1042  3adantl2  1180  3adantr2  1183  enq0tr  7654  ixxssixx  10137  rebtwn2zlemshrink  10514  zsumdc  11950  muldvds1  12382  dvds2add  12391  dvds2sub  12392  dvdstr  12394  pw2dvdslemn  12742  ctinf  13056  mndissubm  13563  gsumfzconst  13933