Theorem simp3bi 999

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
simp3bi	⊢ (𝜑 → 𝜃)

Proof of Theorem simp3bi

Step	Hyp	Ref	Expression
1		3simp1bi.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))
2	1	biimpi 119	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
3	2	simp3d 996	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 965

This theorem is referenced by:  limuni  4351  smores2  6231  ersym  6481  ertr  6484  fvixp  6637  fiintim  6862  eluzle  9430  ef01bndlem  11630  sin01bnd  11631  cos01bnd  11632  sin01gt0  11635  ennnfonelemim  12104  reeff1oleme  13032  cosq14gt0  13092  cosq23lt0  13093  coseq0q4123  13094  coseq00topi  13095  coseq0negpitopi  13096  cosq34lt1  13110  cos02pilt1  13111  ioocosf1o  13114