Theorem simp2bi 1015

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 1012	1 |- ( ph -> ch )

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  4444  smodm  6376  erdm  6629  ixpfn  6790  dif1en  6975  eluzelz  9656  elfz3nn0  10236  ef01bndlem  12038  sin01bnd  12039  cos01bnd  12040  sin01gt0  12044  bitsss  12227  gznegcl  12669  gzcjcl  12670  gzaddcl  12671  gzmulcl  12672  gzabssqcl  12675  4sqlem4a  12685  xpsff1o  13152  subgss  13481  rngmgp  13669  srgmgp  13701  ringmgp  13735  lmodring  14028  lmodprop2d  14081  reeff1oleme  15215  cosq14gt0  15275  cosq23lt0  15276  coseq0q4123  15277  coseq00topi  15278  coseq0negpitopi  15279  cosq34lt1  15293  cos02pilt1  15294  ioocosf1o  15297  gausslemma2dlem1a  15506  2sqlem2  15563  2sqlem3  15565