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  4490  smodm  6448  erdm  6703  ixpfn  6864  dif1en  7054  eluzelz  9748  elfz3nn0  10328  ef01bndlem  12288  sin01bnd  12289  cos01bnd  12290  sin01gt0  12294  bitsss  12477  gznegcl  12919  gzcjcl  12920  gzaddcl  12921  gzmulcl  12922  gzabssqcl  12925  4sqlem4a  12935  xpsff1o  13403  subgss  13732  rngmgp  13920  srgmgp  13952  ringmgp  13986  lmodring  14280  lmodprop2d  14333  reeff1oleme  15467  cosq14gt0  15527  cosq23lt0  15528  coseq0q4123  15529  coseq00topi  15530  coseq0negpitopi  15531  cosq34lt1  15545  cos02pilt1  15546  ioocosf1o  15549  gausslemma2dlem1a  15758  2sqlem2  15815  2sqlem3  15817