Theorem simp3bi 1016

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 1013	1 ⊢ (𝜑 → 𝜃)

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  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  limuni  4427  smores2  6347  ersym  6599  ertr  6602  fvixp  6757  fiintim  6985  eluzle  9604  ef01bndlem  11899  sin01bnd  11900  cos01bnd  11901  sin01gt0  11905  gznegcl  12513  gzcjcl  12514  gzaddcl  12515  gzmulcl  12516  gzabssqcl  12519  4sqlem4a  12529  ennnfonelemim  12581  xpsff1o  12932  subggrp  13247  srgdilem  13465  srgrz  13480  srglz  13481  ringdilem  13508  ringsrg  13543  subrngss  13696  lmodlema  13788  reeff1oleme  14907  cosq14gt0  14967  cosq23lt0  14968  coseq0q4123  14969  coseq00topi  14970  coseq0negpitopi  14971  cosq34lt1  14985  cos02pilt1  14986  ioocosf1o  14989  2sqlem2  15202  2sqlem3  15204