Theorem 3anidm23 1292

Description: Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)

Hypothesis

Ref	Expression
3anidm23.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
3anidm23	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem 3anidm23

Step	Hyp	Ref	Expression
1		3anidm23.1	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜓) → 𝜒)
2	1	3expa 1198	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) → 𝜒)
3	2	anabss3 580	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  efrirr  4338  subeq0  8145  halfaddsub  9112  avglt2  9117  efsub  11644  sinmul  11707  pythagtriplem4  12222  pythagtriplem16  12233  xmet0  13157