Theorem 3anidm13 1429

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 1133	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)
3	2	3anidm12 1428	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093

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 209  df-an 398  df-3an 1095

This theorem is referenced by:  npncan2  11416  ltsubpos  11637  leaddle0  11660  subge02  11661  halfaddsub  12405  avglt1  12410  hashssdif  14369  pythagtriplem4  16785  pythagtriplem14  16794  lsmss2  19637  grpoidinvlem2  30598  hvpncan3  31135  bcm1n  32891  revpfxsfxrev  35359  nnproddivdvdsd  42500  resubidaddlid  42887  reposdif  42960  3anidm12p1  45264  3impcombi  45275