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  4432  smores2  6361  ersym  6613  ertr  6616  fvixp  6771  fiintim  7001  eluzle  9632  ef01bndlem  11940  sin01bnd  11941  cos01bnd  11942  sin01gt0  11946  gznegcl  12571  gzcjcl  12572  gzaddcl  12573  gzmulcl  12574  gzabssqcl  12577  4sqlem4a  12587  ennnfonelemim  12668  prdsbasprj  12986  xpsff1o  13053  subggrp  13385  srgdilem  13603  srgrz  13618  srglz  13619  ringdilem  13646  ringsrg  13681  subrngss  13834  lmodlema  13926  reeff1oleme  15116  cosq14gt0  15176  cosq23lt0  15177  coseq0q4123  15178  coseq00topi  15179  coseq0negpitopi  15180  cosq34lt1  15194  cos02pilt1  15195  ioocosf1o  15198  2sqlem2  15464  2sqlem3  15466