Theorem simp3bi 1038

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 1035	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1002

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 1004

This theorem is referenced by:  limuni  4488  smores2  6451  ersym  6705  ertr  6708  fvixp  6863  en2  6986  fiintim  7109  eluzle  9751  ef01bndlem  12288  sin01bnd  12289  cos01bnd  12290  sin01gt0  12294  gznegcl  12919  gzcjcl  12920  gzaddcl  12921  gzmulcl  12922  gzabssqcl  12925  4sqlem4a  12935  ennnfonelemim  13016  prdsbasprj  13336  xpsff1o  13403  subggrp  13735  srgdilem  13953  srgrz  13968  srglz  13969  ringdilem  13996  ringsrg  14031  subrngss  14185  lmodlema  14277  reeff1oleme  15467  cosq14gt0  15527  cosq23lt0  15528  coseq0q4123  15529  coseq00topi  15530  coseq0negpitopi  15531  cosq34lt1  15545  cos02pilt1  15546  ioocosf1o  15549  2sqlem2  15815  2sqlem3  15817