Theorem simp2bi 962

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
simp2bi	⊢ (𝜑 → 𝜒)

Proof of Theorem simp2bi

Step	Hyp	Ref	Expression
1		3simp1bi.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))
2	1	biimpi 119	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
3	2	simp2d 959	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 927

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

This theorem depends on definitions:  df-bi 116  df-3an 929

This theorem is referenced by:  0ellim  4249  smodm  6094  erdm  6342  ixpfn  6501  dif1en  6675  eluzelz  9127  elfz3nn0  9678  ef01bndlem  11212  sin01bnd  11213  cos01bnd  11214  sin01gt0  11217