Theorem simp2bi 1015

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 1012	1 ⊢ (𝜑 → 𝜒)

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  4430  smodm  6346  erdm  6599  ixpfn  6760  dif1en  6937  eluzelz  9604  elfz3nn0  10184  ef01bndlem  11902  sin01bnd  11903  cos01bnd  11904  sin01gt0  11908  gznegcl  12516  gzcjcl  12517  gzaddcl  12518  gzmulcl  12519  gzabssqcl  12522  4sqlem4a  12532  xpsff1o  12935  subgss  13247  rngmgp  13435  srgmgp  13467  ringmgp  13501  lmodring  13794  lmodprop2d  13847  reeff1oleme  14948  cosq14gt0  15008  cosq23lt0  15009  coseq0q4123  15010  coseq00topi  15011  coseq0negpitopi  15012  cosq34lt1  15026  cos02pilt1  15027  ioocosf1o  15030  gausslemma2dlem1a  15215  2sqlem2  15272  2sqlem3  15274