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  4545  smodm  6566  erdm  6821  ixpfn  6990  dif1en  7187  eluzelz  9947  lincmble  10423  elfz3nn0  10539  ef01bndlem  12548  sin01bnd  12549  cos01bnd  12550  sin01gt0  12554  bitsss  12737  gznegcl  13183  gzcjcl  13184  gzaddcl  13185  gzmulcl  13186  gzabssqcl  13189  4sqlem4a  13199  xpsff1o  13729  subgss  14036  rngmgp  14342  srgmgp  14379  ringmgp  14413  lmodring  14740  lmodprop2d  14794  reeff1oleme  15998  cosq14gt0  16059  cosq23lt0  16060  coseq0q4123  16061  coseq00topi  16062  coseq0negpitopi  16063  cosq34lt1  16078  cos02pilt1  16079  ioocosf1o  16082  gausslemma2dlem1a  16382  2sqlem2  16439  2sqlem3  16441