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  11391  ltsubpos  11612  leaddle0  11635  subge02  11636  halfaddsub  12357  avglt1  12362  hashssdif  14319  pythagtriplem4  16731  pythagtriplem14  16740  lsmss2  19546  grpoidinvlem2  30449  hvpncan3  30986  bcm1n  32739  revpfxsfxrev  35099  nnproddivdvdsd  41983  resubidaddlid  42378  reposdif  42438  3anidm12p1  44789  3impcombi  44800