Theorem example-E3.1b 310

Description: Convert Nested Implication to Nested Conjunction. †

Hypothesis

Ref	Expression
example-E3.1b.1	⊢ (𝜑₁ → (𝜑₂ → (𝜑₃ → (𝜑₄ → (𝜑₅ → 𝜓)))))

Assertion

Ref	Expression
example-E3.1b	⊢ (((((𝜑₁ ∧ 𝜑₂) ∧ 𝜑₃) ∧ 𝜑₄) ∧ 𝜑₅) → 𝜓)

Proof of Theorem example-E3.1b

Step	Hyp	Ref	Expression
1		example-E3.1b.1	. . . . 5 ⊢ (𝜑₁ → (𝜑₂ → (𝜑₃ → (𝜑₄ → (𝜑₅ → 𝜓)))))
2	1	import-P3.34a.RC 306	. . . 4 ⊢ ((𝜑₁ ∧ 𝜑₂) → (𝜑₃ → (𝜑₄ → (𝜑₅ → 𝜓))))
3	2	import-P3.34a.RC 306	. . 3 ⊢ (((𝜑₁ ∧ 𝜑₂) ∧ 𝜑₃) → (𝜑₄ → (𝜑₅ → 𝜓)))
4	3	import-P3.34a.RC 306	. 2 ⊢ ((((𝜑₁ ∧ 𝜑₂) ∧ 𝜑₃) ∧ 𝜑₄) → (𝜑₅ → 𝜓))
5	4	import-P3.34a.RC 306	1 ⊢ (((((𝜑₁ ∧ 𝜑₂) ∧ 𝜑₃) ∧ 𝜑₄) ∧ 𝜑₅) → 𝜓)

Syntax hints:   → wff-imp 10   ∧ wff-and 132

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by: (None)