Theorem simp2bi 1018

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 1015	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 983

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 985

This theorem is referenced by:  0ellim  4466  smodm  6407  erdm  6660  ixpfn  6821  dif1en  7009  eluzelz  9699  elfz3nn0  10279  ef01bndlem  12233  sin01bnd  12234  cos01bnd  12235  sin01gt0  12239  bitsss  12422  gznegcl  12864  gzcjcl  12865  gzaddcl  12866  gzmulcl  12867  gzabssqcl  12870  4sqlem4a  12880  xpsff1o  13348  subgss  13677  rngmgp  13865  srgmgp  13897  ringmgp  13931  lmodring  14224  lmodprop2d  14277  reeff1oleme  15411  cosq14gt0  15471  cosq23lt0  15472  coseq0q4123  15473  coseq00topi  15474  coseq0negpitopi  15475  cosq34lt1  15489  cos02pilt1  15490  ioocosf1o  15493  gausslemma2dlem1a  15702  2sqlem2  15759  2sqlem3  15761