Theorem simp3bi 1041

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
simp3bi	⊢ (𝜑 → 𝜃)

Proof of Theorem simp3bi

Step	Hyp	Ref	Expression
1		3simp1bi.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))
2	1	biimpi 120	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
3	2	simp3d 1038	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1007

This theorem is referenced by:  limuni  4516  smores2  6524  ersym  6778  ertr  6781  fvixp  6937  en2  7064  fiintim  7190  eluzle  9862  lincmble  10333  ef01bndlem  12435  sin01bnd  12436  cos01bnd  12437  sin01gt0  12441  gznegcl  13066  gzcjcl  13067  gzaddcl  13068  gzmulcl  13069  gzabssqcl  13072  4sqlem4a  13082  ennnfonelemim  13164  prdsbasprj  13484  xpsff1o  13551  subggrp  13883  srgdilem  14102  srgrz  14117  srglz  14118  ringdilem  14145  ringsrg  14180  subrngss  14334  lmodlema  14427  reeff1oleme  15624  cosq14gt0  15684  cosq23lt0  15685  coseq0q4123  15686  coseq00topi  15687  coseq0negpitopi  15688  cosq34lt1  15702  cos02pilt1  15703  ioocosf1o  15706  2sqlem2  15975  2sqlem3  15977