Theorem simp1bi 1014

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
simp1bi	⊢ (𝜑 → 𝜓)

Proof of Theorem simp1bi

Step	Hyp	Ref	Expression
1		3simp1bi.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))
2	1	biimpi 120	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
3	2	simp1d 1011	1 ⊢ (𝜑 → 𝜓)

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

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  limord  4431  smores2  6361  smofvon2dm  6363  smofvon  6366  errel  6610  lincmb01cmp  10095  iccf1o  10096  elfznn0  10206  elfzouz  10243  ef01bndlem  11938  sin01bnd  11939  cos01bnd  11940  sin01gt0  11944  cos01gt0  11945  sin02gt0  11946  eulerthlema  12423  modprm0  12448  gzcn  12566  subgbas  13384  subgrcl  13385  rngabl  13567  srgcmn  13598  ringgrp  13633  subrngrcl  13835  lmodgrp  13926  coseq00topi  15155  coseq0negpitopi  15156  cosq34lt1  15170  cos11  15173  nconstwlpolemgt0  15795