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  4487  smores2  6446  ersym  6700  ertr  6703  fvixp  6858  en2  6981  fiintim  7101  eluzle  9742  ef01bndlem  12275  sin01bnd  12276  cos01bnd  12277  sin01gt0  12281  gznegcl  12906  gzcjcl  12907  gzaddcl  12908  gzmulcl  12909  gzabssqcl  12912  4sqlem4a  12922  ennnfonelemim  13003  prdsbasprj  13323  xpsff1o  13390  subggrp  13722  srgdilem  13940  srgrz  13955  srglz  13956  ringdilem  13983  ringsrg  14018  subrngss  14172  lmodlema  14264  reeff1oleme  15454  cosq14gt0  15514  cosq23lt0  15515  coseq0q4123  15516  coseq00topi  15517  coseq0negpitopi  15518  cosq34lt1  15532  cos02pilt1  15533  ioocosf1o  15536  2sqlem2  15802  2sqlem3  15804