Theorem simp1bi 979

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 119	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
3	2	simp1d 976	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 945

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116  df-3an 947

This theorem is referenced by:  limord  4277  smores2  6145  smofvon2dm  6147  smofvon  6150  errel  6392  lincmb01cmp  9679  iccf1o  9680  elfznn0  9787  elfzouz  9821  ef01bndlem  11314  sin01bnd  11315  cos01bnd  11316  sin01gt0  11319  cos01gt0  11320  sin02gt0  11321