Theorem rcp-NDORE4 237

Description: ∨ Elimination Recipe. †

Hypotheses

Ref	Expression
rcp-NDORE4.1	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝜑) → 𝜒)
rcp-NDORE4.2	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝜓) → 𝜒)
rcp-NDORE4.3	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → (𝜑 ∨ 𝜓))

Assertion

Ref	Expression
rcp-NDORE4	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜒)

Proof of Theorem rcp-NDORE4

Step	Hyp	Ref	Expression
1		rcp-NDORE4.1	. . 3 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝜑) → 𝜒)
2	1	rcp-NDSEP4 187	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) ∧ 𝜑) → 𝜒)
3		rcp-NDORE4.2	. . 3 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃ ∧ 𝜓) → 𝜒)
4	3	rcp-NDSEP4 187	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) ∧ 𝜓) → 𝜒)
5		rcp-NDORE4.3	. 2 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → (𝜑 ∨ 𝜓))
6	2, 4, 5	ndore-P3.12 177	1 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝛾₃) → 𝜒)

Syntax hints:   → wff-imp 10   ∨ wff-or 144   ∧ wff-rcp-AND3 160   ∧ wff-rcp-AND4 162

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-or-D2.3 145  df-rcp-AND4 163

This theorem is referenced by:  ndore-P3.12.CL  247  oroverand-P4.27-L4  463