Theorem simp3bi 1045

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 1042	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 1009

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 1011

This theorem is referenced by:  limuni  4539  smores2  6559  ersym  6813  ertr  6816  fvixp  6979  en2  7106  fiintim  7232  eluzle  9917  lincmble  10389  ef01bndlem  12506  sin01bnd  12507  cos01bnd  12508  sin01gt0  12512  gznegcl  13137  gzcjcl  13138  gzaddcl  13139  gzmulcl  13140  gzabssqcl  13143  4sqlem4a  13153  ennnfonelemim  13298  xpsff1o  13653  subggrp  13963  prdsbasprj  14165  srgdilem  14256  srgrz  14271  srglz  14272  ringdilem  14299  ringsrg  14335  subrngss  14491  lmodlema  14611  reeff1oleme  15856  cosq14gt0  15916  cosq23lt0  15917  coseq0q4123  15918  coseq00topi  15919  coseq0negpitopi  15920  cosq34lt1  15934  cos02pilt1  15935  ioocosf1o  15938  2sqlem2  16217  2sqlem3  16219