Theorem simp2bi 1039

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 1036	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1004

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 1006

This theorem is referenced by:  0ellim  4495  smodm  6457  erdm  6712  ixpfn  6873  dif1en  7068  eluzelz  9765  elfz3nn0  10350  ef01bndlem  12319  sin01bnd  12320  cos01bnd  12321  sin01gt0  12325  bitsss  12508  gznegcl  12950  gzcjcl  12951  gzaddcl  12952  gzmulcl  12953  gzabssqcl  12956  4sqlem4a  12966  xpsff1o  13434  subgss  13763  rngmgp  13952  srgmgp  13984  ringmgp  14018  lmodring  14312  lmodprop2d  14365  reeff1oleme  15499  cosq14gt0  15559  cosq23lt0  15560  coseq0q4123  15561  coseq00topi  15562  coseq0negpitopi  15563  cosq34lt1  15577  cos02pilt1  15578  ioocosf1o  15581  gausslemma2dlem1a  15790  2sqlem2  15847  2sqlem3  15849