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  11515  ltsubpos  11734  leaddle0  11757  subge02  11758  halfaddsub  12479  avglt1  12484  hashssdif  14435  pythagtriplem4  16844  pythagtriplem14  16853  lsmss2  19653  grpoidinvlem2  30491  hvpncan3  31028  bcm1n  32777  revpfxsfxrev  35143  nnproddivdvdsd  42018  resubidaddlid  42413  reposdif  42461  3anidm12p1  44805  3impcombi  44816