Theorem 3anidm12 1420

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 1123	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))
3	2	anabsi5 668	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088

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 206  df-an 398  df-3an 1090

This theorem is referenced by:  3anidm13  1421  syl2an3an  1423  dedth3v  4592  fovcl  7537  nncan  11489  divid  11901  sqdivid  14087  subsq  14174  o1lo1  15481  retancl  16085  tanneg  16091  gcd0id  16460  coprm  16648  ablonncan  29809  kbpj  31209  xdivid  32094  xrsmulgzz  32179  f1resrcmplf1dlem  34089  expgrowthi  43092  dvconstbi  43093  3ornot23  43270  3anidm12p2  43568  sinhpcosh  47785  reseccl  47798  recsccl  47799  recotcl  47800  onetansqsecsq  47806