Theorem simp2bi 1015

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 120	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
3	2	simp2d 1012	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 980

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

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  0ellim  4433  smodm  6349  erdm  6602  ixpfn  6763  dif1en  6940  eluzelz  9610  elfz3nn0  10190  ef01bndlem  11921  sin01bnd  11922  cos01bnd  11923  sin01gt0  11927  bitsss  12110  gznegcl  12544  gzcjcl  12545  gzaddcl  12546  gzmulcl  12547  gzabssqcl  12550  4sqlem4a  12560  xpsff1o  12992  subgss  13304  rngmgp  13492  srgmgp  13524  ringmgp  13558  lmodring  13851  lmodprop2d  13904  reeff1oleme  15008  cosq14gt0  15068  cosq23lt0  15069  coseq0q4123  15070  coseq00topi  15071  coseq0negpitopi  15072  cosq34lt1  15086  cos02pilt1  15087  ioocosf1o  15090  gausslemma2dlem1a  15299  2sqlem2  15356  2sqlem3  15358