Theorem simp1bi 954

Description: Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypothesis

Ref	Expression
3simp1bi.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

Assertion

Ref	Expression
simp1bi	⊢ (𝜑 → 𝜓)

Proof of Theorem simp1bi

Step	Hyp	Ref	Expression
1		3simp1bi.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))
2	1	biimpi 118	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
3	2	simp1d 951	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 103   ∧ w3a 920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem depends on definitions:  df-bi 115  df-3an 922

This theorem is referenced by:  limord  4178  smores2  5964  smofvon2dm  5966  smofvon  5969  errel  6203  lincmb01cmp  9171  iccf1o  9172  elfznn0  9277  elfzouz  9308