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  11422  ltsubpos  11643  leaddle0  11666  subge02  11667  halfaddsub  12388  avglt1  12393  hashssdif  14349  pythagtriplem4  16761  pythagtriplem14  16770  lsmss2  19613  grpoidinvlem2  30599  hvpncan3  31136  bcm1n  32892  revpfxsfxrev  35338  nnproddivdvdsd  42399  resubidaddlid  42794  reposdif  42854  3anidm12p1  45190  3impcombi  45201