Theorem 3anidm13 1445

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 1142	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)
3	2	3anidm12 1444	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101

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 210  df-an 401  df-3an 1103

This theorem is referenced by:  npncan2  11488  ltsubpos  11709  leaddle0  11732  subge02  11733  halfaddsub  12480  avglt1  12485  hashssdif  14452  pythagtriplem4  16882  pythagtriplem14  16891  lsmss2  19740  grpoidinvlem2  30827  hvpncan3  31364  bcm1n  33110  revpfxsfxrev  35565  nnproddivdvdsd  42717  resubidaddlid  43106  reposdif  43179  3anidm12p1  45466  3impcombi  45477