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  11416  ltsubpos  11637  leaddle0  11660  subge02  11661  halfaddsub  12405  avglt1  12410  hashssdif  14369  pythagtriplem4  16785  pythagtriplem14  16794  lsmss2  19637  grpoidinvlem2  30595  hvpncan3  31132  bcm1n  32887  revpfxsfxrev  35318  nnproddivdvdsd  42459  resubidaddlid  42847  reposdif  42920  3anidm12p1  45256  3impcombi  45267