Theorem simp2bi 1040

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 1037	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1005

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 1007

This theorem is referenced by:  0ellim  4518  smodm  6521  erdm  6776  ixpfn  6938  dif1en  7135  eluzelz  9859  lincmble  10333  elfz3nn0  10445  ef01bndlem  12435  sin01bnd  12436  cos01bnd  12437  sin01gt0  12441  bitsss  12624  gznegcl  13066  gzcjcl  13067  gzaddcl  13068  gzmulcl  13069  gzabssqcl  13072  4sqlem4a  13082  xpsff1o  13551  subgss  13880  rngmgp  14069  srgmgp  14101  ringmgp  14135  lmodring  14430  lmodprop2d  14483  reeff1oleme  15624  cosq14gt0  15684  cosq23lt0  15685  coseq0q4123  15686  coseq00topi  15687  coseq0negpitopi  15688  cosq34lt1  15702  cos02pilt1  15703  ioocosf1o  15706  gausslemma2dlem1a  15918  2sqlem2  15975  2sqlem3  15977