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  4538  fovcl  7480  nncan  11396  divid  11813  dividOLD  11814  sqdivid  14035  subsq  14123  o1lo1  15450  retancl  16057  tanneg  16063  gcd0id  16436  coprm  16628  ablonncan  30543  kbpj  31943  xdivid  32915  xrsmulgzz  32997  f1resrcmplf1dlem  35105  expgrowthi  44431  dvconstbi  44432  3ornot23  44607  3anidm12p2  44904  sinhpcosh  49846  reseccl  49859  recsccl  49860  recotcl  49861  onetansqsecsq  49867