Theorem example-E3.2b 312

Description: Convert Nested Implication to Sequent. †

Hypothesis

Ref	Expression
example-E3.2b.1	⊢ (𝜑₁ → (𝜑₂ → (𝜑₃ → (𝜑₄ → (𝜑₅ → 𝜓)))))

Assertion

Ref	Expression
example-E3.2b	⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃ ∧ 𝜑₄ ∧ 𝜑₅) → 𝜓)

Proof of Theorem example-E3.2b

Step	Hyp	Ref	Expression
1		rcp-NDASM5of5 206	. 2 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃ ∧ 𝜑₄ ∧ 𝜑₅) → 𝜑₅)
2		rcp-NDASM4of4 201	. . . 4 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃ ∧ 𝜑₄) → 𝜑₄)
3		rcp-NDASM3of3 197	. . . . . 6 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃) → 𝜑₃)
4		rcp-NDASM2of2 194	. . . . . . . 8 ⊢ ((𝜑₁ ∧ 𝜑₂) → 𝜑₂)
5		example-E3.2b.1	. . . . . . . . 9 ⊢ (𝜑₁ → (𝜑₂ → (𝜑₃ → (𝜑₄ → (𝜑₅ → 𝜓)))))
6	5	rcp-NDIMP1add1 208	. . . . . . . 8 ⊢ ((𝜑₁ ∧ 𝜑₂) → (𝜑₂ → (𝜑₃ → (𝜑₄ → (𝜑₅ → 𝜓)))))
7	4, 6	ndime-P3.6 171	. . . . . . 7 ⊢ ((𝜑₁ ∧ 𝜑₂) → (𝜑₃ → (𝜑₄ → (𝜑₅ → 𝜓))))
8	7	rcp-NDIMP2add1 209	. . . . . 6 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃) → (𝜑₃ → (𝜑₄ → (𝜑₅ → 𝜓))))
9	3, 8	ndime-P3.6 171	. . . . 5 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃) → (𝜑₄ → (𝜑₅ → 𝜓)))
10	9	rcp-NDIMP3add1 210	. . . 4 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃ ∧ 𝜑₄) → (𝜑₄ → (𝜑₅ → 𝜓)))
11	2, 10	ndime-P3.6 171	. . 3 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃ ∧ 𝜑₄) → (𝜑₅ → 𝜓))
12	11	rcp-NDIMP4add1 211	. 2 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃ ∧ 𝜑₄ ∧ 𝜑₅) → (𝜑₅ → 𝜓))
13	1, 12	ndime-P3.6 171	1 ⊢ ((𝜑₁ ∧ 𝜑₂ ∧ 𝜑₃ ∧ 𝜑₄ ∧ 𝜑₅) → 𝜓)

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∧ wff-rcp-AND3 160   ∧ wff-rcp-AND4 162   ∧ wff-rcp-AND5 164

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-rcp-AND3 161  df-rcp-AND4 163  df-rcp-AND5 165

This theorem is referenced by: (None)