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  4590  fovcl  7532  nncan  11485  divid  11897  sqdivid  14083  subsq  14170  o1lo1  15477  retancl  16081  tanneg  16087  gcd0id  16456  coprm  16644  ablonncan  29787  kbpj  31187  xdivid  32072  xrsmulgzz  32157  f1resrcmplf1dlem  34027  expgrowthi  43025  dvconstbi  43026  3ornot23  43203  3anidm12p2  43501  sinhpcosh  47687  reseccl  47700  recsccl  47701  recotcl  47702  onetansqsecsq  47708