Theorem 3anidm12 1418

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 1121	. 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  1419  syl2an3an  1421  dedth3v  4594  fovcl  7561  nncan  11536  divid  11951  dividOLD  11952  sqdivid  14159  subsq  14246  o1lo1  15570  retancl  16175  tanneg  16181  gcd0id  16553  coprm  16745  ablonncan  30585  kbpj  31985  xdivid  32895  xrsmulgzz  32994  f1resrcmplf1dlem  35079  expgrowthi  44329  dvconstbi  44330  3ornot23  44507  3anidm12p2  44805  sinhpcosh  48971  reseccl  48984  recsccl  48985  recotcl  48986  onetansqsecsq  48992