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  11449  ltsubpos  11670  leaddle0  11693  subge02  11694  halfaddsub  12415  avglt1  12420  hashssdif  14377  pythagtriplem4  16790  pythagtriplem14  16799  lsmss2  19597  grpoidinvlem2  30434  hvpncan3  30971  bcm1n  32718  revpfxsfxrev  35103  nnproddivdvdsd  41988  resubidaddlid  42383  reposdif  42443  3anidm12p1  44795  3impcombi  44806