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  9630  ef01bndlem  11938  sin01bnd  11939  cos01bnd  11940  sin01gt0  11944  gznegcl  12569  gzcjcl  12570  gzaddcl  12571  gzmulcl  12572  gzabssqcl  12575  4sqlem4a  12585  ennnfonelemim  12666  prdsbasprj  12984  xpsff1o  13051  subggrp  13383  srgdilem  13601  srgrz  13616  srglz  13617  ringdilem  13644  ringsrg  13679  subrngss  13832  lmodlema  13924  reeff1oleme  15092  cosq14gt0  15152  cosq23lt0  15153  coseq0q4123  15154  coseq00topi  15155  coseq0negpitopi  15156  cosq34lt1  15170  cos02pilt1  15171  ioocosf1o  15174  2sqlem2  15440  2sqlem3  15442