Theorem 3anidm12 1238

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 1149	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))
3	2	anabsi5 547	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 927

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 929

This theorem is referenced by:  3anidm13  1239  syl2an3an  1241  prarloclemarch2  7075  nq02m  7121  recexprlem1ssl  7289  recexprlem1ssu  7290  nncan  7808  dividap  8265  modqid0  9906  subsq  10176  retanclap  11162  tannegap  11168  gcd0id  11397  coprm  11550