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  4429  smodm  6344  erdm  6597  ixpfn  6758  dif1en  6935  eluzelz  9601  elfz3nn0  10181  ef01bndlem  11899  sin01bnd  11900  cos01bnd  11901  sin01gt0  11905  gznegcl  12513  gzcjcl  12514  gzaddcl  12515  gzmulcl  12516  gzabssqcl  12519  4sqlem4a  12529  xpsff1o  12932  subgss  13244  rngmgp  13432  srgmgp  13464  ringmgp  13498  lmodring  13791  lmodprop2d  13844  reeff1oleme  14907  cosq14gt0  14967  cosq23lt0  14968  coseq0q4123  14969  coseq00topi  14970  coseq0negpitopi  14971  cosq34lt1  14985  cos02pilt1  14986  ioocosf1o  14989  gausslemma2dlem1a  15174  2sqlem2  15202  2sqlem3  15204