Theorem simp3bi 1019

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 1016	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 983

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 985

This theorem is referenced by:  limuni  4464  smores2  6410  ersym  6662  ertr  6665  fvixp  6820  en2  6943  fiintim  7061  eluzle  9702  ef01bndlem  12233  sin01bnd  12234  cos01bnd  12235  sin01gt0  12239  gznegcl  12864  gzcjcl  12865  gzaddcl  12866  gzmulcl  12867  gzabssqcl  12870  4sqlem4a  12880  ennnfonelemim  12961  prdsbasprj  13281  xpsff1o  13348  subggrp  13680  srgdilem  13898  srgrz  13913  srglz  13914  ringdilem  13941  ringsrg  13976  subrngss  14129  lmodlema  14221  reeff1oleme  15411  cosq14gt0  15471  cosq23lt0  15472  coseq0q4123  15473  coseq00topi  15474  coseq0negpitopi  15475  cosq34lt1  15489  cos02pilt1  15490  ioocosf1o  15493  2sqlem2  15759  2sqlem3  15761