Theorem simp2bi 1044

Description: Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypothesis

Ref	Expression
3simp1bi.1	|- ( ph <-> ( ps /\ ch /\ th ) )

Assertion

Ref	Expression
simp2bi	|- ( ph -> ch )

Proof of Theorem simp2bi

Step	Hyp	Ref	Expression
1		3simp1bi.1	. . 3 |- ( ph <-> ( ps /\ ch /\ th ) )
2	1	biimpi 120	. 2 |- ( ph -> ( ps /\ ch /\ th ) )
3	2	simp2d 1041	1 |- ( ph -> ch )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  0ellim  4543  smodm  6562  erdm  6817  ixpfn  6986  dif1en  7183  eluzelz  9931  lincmble  10406  elfz3nn0  10522  ef01bndlem  12523  sin01bnd  12524  cos01bnd  12525  sin01gt0  12529  bitsss  12712  gznegcl  13154  gzcjcl  13155  gzaddcl  13156  gzmulcl  13157  gzabssqcl  13160  4sqlem4a  13170  xpsff1o  13670  subgss  13977  rngmgp  14235  srgmgp  14272  ringmgp  14306  lmodring  14631  lmodprop2d  14685  reeff1oleme  15873  cosq14gt0  15933  cosq23lt0  15934  coseq0q4123  15935  coseq00topi  15936  coseq0negpitopi  15937  cosq34lt1  15951  cos02pilt1  15952  ioocosf1o  15955  gausslemma2dlem1a  16177  2sqlem2  16234  2sqlem3  16236