Theorem 3anidm23 1338

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 1234	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) → 𝜒)
3	2	anabss3 591	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  efrirr  4498  subeq0  8552  halfaddsub  9541  avglt2  9547  efsub  12450  sinmul  12513  pythagtriplem4  13049  pythagtriplem16  13060  ballotfilemfc0  13234  ballotfilemfcc  13235  xmet0  15466