Theorem simp2bi 1003

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 119	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
3	2	simp2d 1000	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by:  0ellim  4376  smodm  6259  erdm  6511  ixpfn  6670  dif1en  6845  eluzelz  9475  elfz3nn0  10050  ef01bndlem  11697  sin01bnd  11698  cos01bnd  11699  sin01gt0  11702  gznegcl  12305  gzcjcl  12306  gzaddcl  12307  gzmulcl  12308  gzabssqcl  12311  4sqlem4a  12321  reeff1oleme  13333  cosq14gt0  13393  cosq23lt0  13394  coseq0q4123  13395  coseq00topi  13396  coseq0negpitopi  13397  cosq34lt1  13411  cos02pilt1  13412  ioocosf1o  13415  2sqlem2  13591  2sqlem3  13593