This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem naim2

Description: Constructor theorem for -/\ . (Contributed by Anthony Hart, 1-Sep-2011)

Ref Expression
Assertion naim2 ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 ⊼ 𝜓 ) → ( 𝜒 ⊼ 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 con3 ⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜑 ) )
2 1 orim2d ⊢ ( ( 𝜑 → 𝜓 ) → ( ( ¬ 𝜒 ∨ ¬ 𝜓 ) → ( ¬ 𝜒 ∨ ¬ 𝜑 ) ) )
3 pm3.13 ⊢ ( ¬ ( 𝜒 ∧ 𝜓 ) → ( ¬ 𝜒 ∨ ¬ 𝜓 ) )
4 pm3.14 ⊢ ( ( ¬ 𝜒 ∨ ¬ 𝜑 ) → ¬ ( 𝜒 ∧ 𝜑 ) )
5 3 4 imim12i ⊢ ( ( ( ¬ 𝜒 ∨ ¬ 𝜓 ) → ( ¬ 𝜒 ∨ ¬ 𝜑 ) ) → ( ¬ ( 𝜒 ∧ 𝜓 ) → ¬ ( 𝜒 ∧ 𝜑 ) ) )
6 df-nan ⊢ ( ( 𝜒 ⊼ 𝜓 ) ↔ ¬ ( 𝜒 ∧ 𝜓 ) )
7 df-nan ⊢ ( ( 𝜒 ⊼ 𝜑 ) ↔ ¬ ( 𝜒 ∧ 𝜑 ) )
8 5 6 7 3imtr4g ⊢ ( ( ( ¬ 𝜒 ∨ ¬ 𝜓 ) → ( ¬ 𝜒 ∨ ¬ 𝜑 ) ) → ( ( 𝜒 ⊼ 𝜓 ) → ( 𝜒 ⊼ 𝜑 ) ) )
9 2 8 syl ⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 ⊼ 𝜓 ) → ( 𝜒 ⊼ 𝜑 ) ) )