Theorem simp2bi 1044

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 1041	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1009

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 1011

This theorem is referenced by:  0ellim  4541  smodm  6556  erdm  6811  ixpfn  6980  dif1en  7177  eluzelz  9914  lincmble  10389  elfz3nn0  10505  ef01bndlem  12506  sin01bnd  12507  cos01bnd  12508  sin01gt0  12512  bitsss  12695  gznegcl  13137  gzcjcl  13138  gzaddcl  13139  gzmulcl  13140  gzabssqcl  13143  4sqlem4a  13153  xpsff1o  13653  subgss  13960  rngmgp  14218  srgmgp  14255  ringmgp  14289  lmodring  14614  lmodprop2d  14668  reeff1oleme  15856  cosq14gt0  15916  cosq23lt0  15917  coseq0q4123  15918  coseq00topi  15919  coseq0negpitopi  15920  cosq34lt1  15934  cos02pilt1  15935  ioocosf1o  15938  gausslemma2dlem1a  16160  2sqlem2  16217  2sqlem3  16219