Theorem 3simpb 947

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 938	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓))
2		3simpa 946	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → (𝜑 ∧ 𝜒))
3	1, 2	sylbi 120	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 930

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  df-3an 932

This theorem is referenced by:  3adant2  968  3adantl2  1106  3adantr2  1109  enq0tr  7143  ixxssixx  9526  rebtwn2zlemshrink  9872  zsumdc  10992  muldvds1  11313  dvds2add  11322  dvds2sub  11323  dvdstr  11325  pw2dvdslemn  11635  ctinf  11735