Theorem naim1

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

	Ref	Expression
Assertion	naim1	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 ⊼ 𝜒 ) → ( 𝜑 ⊼ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		con3	⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜑 ) )
2	1	orim1d	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ¬ 𝜓 ∨ ¬ 𝜒 ) → ( ¬ 𝜑 ∨ ¬ 𝜒 ) ) )
3		pm3.13	⊢ ( ¬ ( 𝜓 ∧ 𝜒 ) → ( ¬ 𝜓 ∨ ¬ 𝜒 ) )
4		pm3.14	⊢ ( ( ¬ 𝜑 ∨ ¬ 𝜒 ) → ¬ ( 𝜑 ∧ 𝜒 ) )
5	3 4	imim12i	⊢ ( ( ( ¬ 𝜓 ∨ ¬ 𝜒 ) → ( ¬ 𝜑 ∨ ¬ 𝜒 ) ) → ( ¬ ( 𝜓 ∧ 𝜒 ) → ¬ ( 𝜑 ∧ 𝜒 ) ) )
6		df-nan	⊢ ( ( 𝜓 ⊼ 𝜒 ) ↔ ¬ ( 𝜓 ∧ 𝜒 ) )
7		df-nan	⊢ ( ( 𝜑 ⊼ 𝜒 ) ↔ ¬ ( 𝜑 ∧ 𝜒 ) )
8	5 6 7	3imtr4g	⊢ ( ( ( ¬ 𝜓 ∨ ¬ 𝜒 ) → ( ¬ 𝜑 ∨ ¬ 𝜒 ) ) → ( ( 𝜓 ⊼ 𝜒 ) → ( 𝜑 ⊼ 𝜒 ) ) )
9	2 8	syl	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 ⊼ 𝜒 ) → ( 𝜑 ⊼ 𝜒 ) ) )