Theorem 3anidm13 1419

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 1125	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)
3	2	3anidm12 1418	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 396   ∧ 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 206  df-an 397  df-3an 1088

This theorem is referenced by:  npncan2  11248  ltsubpos  11467  leaddle0  11490  subge02  11491  halfaddsub  12206  avglt1  12211  hashssdif  14127  pythagtriplem4  16520  pythagtriplem14  16529  lsmss2  19273  grpoidinvlem2  28867  hvpncan3  29404  bcm1n  31116  revpfxsfxrev  33077  nnproddivdvdsd  40009  resubidaddid1  40378  reposdif  40424  3anidm12p1  42426  3impcombi  42437