Theorem simp2bi 1039

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 1036	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1004

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 1006

This theorem is referenced by:  0ellim  4497  smodm  6462  erdm  6717  ixpfn  6878  dif1en  7073  eluzelz  9770  elfz3nn0  10355  ef01bndlem  12340  sin01bnd  12341  cos01bnd  12342  sin01gt0  12346  bitsss  12529  gznegcl  12971  gzcjcl  12972  gzaddcl  12973  gzmulcl  12974  gzabssqcl  12977  4sqlem4a  12987  xpsff1o  13455  subgss  13784  rngmgp  13973  srgmgp  14005  ringmgp  14039  lmodring  14333  lmodprop2d  14386  reeff1oleme  15525  cosq14gt0  15585  cosq23lt0  15586  coseq0q4123  15587  coseq00topi  15588  coseq0negpitopi  15589  cosq34lt1  15603  cos02pilt1  15604  ioocosf1o  15607  gausslemma2dlem1a  15816  2sqlem2  15873  2sqlem3  15875