Theorem animpimp2impd

Description: Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022)

	Ref	Expression
Hypotheses	animpimp2impd.1	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝜒 → ( 𝜃 → 𝜂 ) ) )
	animpimp2impd.2	⊢ ( ( 𝜓 ∧ ( 𝜑 ∧ 𝜃 ) ) → ( 𝜂 → 𝜏 ) )
Assertion	animpimp2impd	⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜏 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		animpimp2impd.1	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝜒 → ( 𝜃 → 𝜂 ) ) )
2		animpimp2impd.2	⊢ ( ( 𝜓 ∧ ( 𝜑 ∧ 𝜃 ) ) → ( 𝜂 → 𝜏 ) )
3	2	expr	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝜃 → ( 𝜂 → 𝜏 ) ) )
4	3	a2d	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( ( 𝜃 → 𝜂 ) → ( 𝜃 → 𝜏 ) ) )
5	1 4	syld	⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝜒 → ( 𝜃 → 𝜏 ) ) )
6	5	expcom	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) ) )
7	6	a2d	⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜏 ) ) ) )