Theorem 3anidm13 1441

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 1140	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)
3	2	3anidm12 1440	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099

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 400  df-3an 1101

This theorem is referenced by:  npncan2  11460  ltsubpos  11681  leaddle0  11704  subge02  11705  halfaddsub  12456  avglt1  12461  hashssdif  14427  pythagtriplem4  16857  pythagtriplem14  16866  lsmss2  19709  grpoidinvlem2  30710  hvpncan3  31247  bcm1n  32999  revpfxsfxrev  35470  nnproddivdvdsd  42622  resubidaddlid  43009  reposdif  43082  3anidm12p1  45386  3impcombi  45397