Theorem simp3bi 1040

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 1037	1 ⊢ (𝜑 → 𝜃)

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  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1006

This theorem is referenced by:  limuni  4495  smores2  6465  ersym  6719  ertr  6722  fvixp  6877  en2  7003  fiintim  7128  eluzle  9773  ef01bndlem  12340  sin01bnd  12341  cos01bnd  12342  sin01gt0  12346  gznegcl  12971  gzcjcl  12972  gzaddcl  12973  gzmulcl  12974  gzabssqcl  12977  4sqlem4a  12987  ennnfonelemim  13068  prdsbasprj  13388  xpsff1o  13455  subggrp  13787  srgdilem  14006  srgrz  14021  srglz  14022  ringdilem  14049  ringsrg  14084  subrngss  14238  lmodlema  14330  reeff1oleme  15525  cosq14gt0  15585  cosq23lt0  15586  coseq0q4123  15587  coseq00topi  15588  coseq0negpitopi  15589  cosq34lt1  15603  cos02pilt1  15604  ioocosf1o  15607  2sqlem2  15873  2sqlem3  15875