Theorem rcp-NDORE3 236

Description: ∨ Elimination Recipe. †

Hypotheses

Ref	Expression
rcp-NDORE3.1	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝜑) → 𝜒)
rcp-NDORE3.2	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝜓) → 𝜒)
rcp-NDORE3.3	⊢ ((𝛾₁ ∧ 𝛾₂) → (𝜑 ∨ 𝜓))

Assertion

Ref	Expression
rcp-NDORE3	⊢ ((𝛾₁ ∧ 𝛾₂) → 𝜒)

Proof of Theorem rcp-NDORE3

Step	Hyp	Ref	Expression
1		rcp-NDORE3.1	. . 3 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝜑) → 𝜒)
2	1	rcp-NDSEP3 186	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝜑) → 𝜒)
3		rcp-NDORE3.2	. . 3 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ 𝜓) → 𝜒)
4	3	rcp-NDSEP3 186	. 2 ⊢ (((𝛾₁ ∧ 𝛾₂) ∧ 𝜓) → 𝜒)
5		rcp-NDORE3.3	. 2 ⊢ ((𝛾₁ ∧ 𝛾₂) → (𝜑 ∨ 𝜓))
6	2, 4, 5	ndore-P3.12 177	1 ⊢ ((𝛾₁ ∧ 𝛾₂) → 𝜒)

Syntax hints:   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144   ∧ wff-rcp-AND3 160

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-AND3 161

This theorem is referenced by:  orassoc-P3.38-L1  320  orassoc-P3.38-L2  321  suborl-P3.43a-L1  345  orasim-P3.48-L1  359  joinimandinc-P4.8a  397  joinimor-P4.8c  403  dmorgarev-L4.2  453  andoveror-P4.27-L1  459  oroverand-P4.27-L4  463