Theorem 3anidm13 1423

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 1127	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜓) → 𝜒)
3	2	3anidm12 1422	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 1089

This theorem is referenced by:  npncan2  11412  ltsubpos  11633  leaddle0  11656  subge02  11657  halfaddsub  12378  avglt1  12383  hashssdif  14339  pythagtriplem4  16751  pythagtriplem14  16760  lsmss2  19600  grpoidinvlem2  30584  hvpncan3  31121  bcm1n  32877  revpfxsfxrev  35312  nnproddivdvdsd  42322  resubidaddlid  42717  reposdif  42777  3anidm12p1  45113  3impcombi  45124