Theorem 3anidm13 1422

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 1126	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)
3	2	3anidm12 1421	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:  npncan2  11406  ltsubpos  11627  leaddle0  11650  subge02  11651  halfaddsub  12372  avglt1  12377  hashssdif  14333  pythagtriplem4  16745  pythagtriplem14  16754  lsmss2  19594  grpoidinvlem2  30529  hvpncan3  31066  bcm1n  32824  revpfxsfxrev  35259  nnproddivdvdsd  42193  resubidaddlid  42592  reposdif  42652  3anidm12p1  44988  3impcombi  44999