Theorem 3anidm13 1423

Description: Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)

Hypothesis

Ref	Expression
3anidm13.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒)

Assertion

Ref	Expression
3anidm13	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem 3anidm13

Step	Hyp	Ref	Expression
1		3anidm13.1	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → 𝜒)
2	1	3com23 1127	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)
3	2	3anidm12 1422	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 1089

This theorem is referenced by:  npncan2  11421  ltsubpos  11642  leaddle0  11665  subge02  11666  halfaddsub  12410  avglt1  12415  hashssdif  14374  pythagtriplem4  16790  pythagtriplem14  16799  lsmss2  19642  grpoidinvlem2  30576  hvpncan3  31113  bcm1n  32868  revpfxsfxrev  35298  nnproddivdvdsd  42439  resubidaddlid  42827  reposdif  42900  3anidm12p1  45232  3impcombi  45243