Theorem 3anidm12 1421

Description: Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)

Hypothesis

Ref	Expression
3anidm12.1	⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
3anidm12	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem 3anidm12

Step	Hyp	Ref	Expression
1		3anidm12.1	. . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)
2	1	3expib 1122	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))
3	2	anabsi5 669	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088

This theorem is referenced by:  3anidm13  1422  syl2an3an  1424  dedth3v  4541  fovcl  7484  nncan  11408  divid  11825  dividOLD  11826  sqdivid  14043  subsq  14131  o1lo1  15458  retancl  16065  tanneg  16071  gcd0id  16444  coprm  16636  ablonncan  30580  kbpj  31980  xdivid  32958  xrsmulgzz  33040  f1resrcmplf1dlem  35191  expgrowthi  44516  dvconstbi  44517  3ornot23  44692  3anidm12p2  44989  sinhpcosh  49927  reseccl  49940  recsccl  49941  recotcl  49942  onetansqsecsq  49948