Theorem 3anidm12 1306

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 1208	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))
3	2	anabsi5 579	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  3anidm13  1307  syl2an3an  1309  fovcl  6025  prarloclemarch2  7481  nq02m  7527  recexprlem1ssl  7695  recexprlem1ssu  7696  nncan  8250  dividap  8722  modqid0  10424  sqdividap  10678  subsq  10720  retanclap  11868  tannegap  11874  gcd0id  12119  coprm  12285