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  11408  ltsubpos  11629  leaddle0  11652  subge02  11653  halfaddsub  12374  avglt1  12379  hashssdif  14335  pythagtriplem4  16747  pythagtriplem14  16756  lsmss2  19596  grpoidinvlem2  30580  hvpncan3  31117  bcm1n  32875  revpfxsfxrev  35310  nnproddivdvdsd  42250  resubidaddlid  42646  reposdif  42706  3anidm12p1  45042  3impcombi  45053