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 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  1421  syl2an3an  1423  dedth3v  4588  fovcl  7562  nncan  11539  divid  11954  dividOLD  11955  sqdivid  14163  subsq  14250  o1lo1  15574  retancl  16179  tanneg  16185  gcd0id  16557  coprm  16749  ablonncan  30576  kbpj  31976  xdivid  32911  xrsmulgzz  33012  f1resrcmplf1dlem  35101  expgrowthi  44357  dvconstbi  44358  3ornot23  44534  3anidm12p2  44832  sinhpcosh  49314  reseccl  49327  recsccl  49328  recotcl  49329  onetansqsecsq  49335