Theorem simp2bi 1037

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 1034	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1002

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 1004

This theorem is referenced by:  0ellim  4493  smodm  6452  erdm  6707  ixpfn  6868  dif1en  7063  eluzelz  9758  elfz3nn0  10343  ef01bndlem  12310  sin01bnd  12311  cos01bnd  12312  sin01gt0  12316  bitsss  12499  gznegcl  12941  gzcjcl  12942  gzaddcl  12943  gzmulcl  12944  gzabssqcl  12947  4sqlem4a  12957  xpsff1o  13425  subgss  13754  rngmgp  13942  srgmgp  13974  ringmgp  14008  lmodring  14302  lmodprop2d  14355  reeff1oleme  15489  cosq14gt0  15549  cosq23lt0  15550  coseq0q4123  15551  coseq00topi  15552  coseq0negpitopi  15553  cosq34lt1  15567  cos02pilt1  15568  ioocosf1o  15571  gausslemma2dlem1a  15780  2sqlem2  15837  2sqlem3  15839