Theorem rcp-RAA3 517

Description: Reductio ad Absurdum.

Hypotheses

Ref	Expression
rcp-RAA3.1	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ ¬ 𝛾₃) → 𝜑)
rcp-RAA3.2	⊢ ((𝛾₁ ∧ 𝛾₂ ∧ ¬ 𝛾₃) → ¬ 𝜑)

Assertion

Ref	Expression
rcp-RAA3	⊢ ((𝛾₁ ∧ 𝛾₂) → 𝛾₃)

Proof of Theorem rcp-RAA3

Step	Hyp	Ref	Expression
1		rcp-RAA3.1	. . 3 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ ¬ 𝛾₃) → 𝜑)
2		rcp-RAA3.2	. . 3 ⊢ ((𝛾₁ ∧ 𝛾₂ ∧ ¬ 𝛾₃) → ¬ 𝜑)
3	1, 2	rcp-NDNEGI3 220	. 2 ⊢ ((𝛾₁ ∧ 𝛾₂) → ¬ ¬ 𝛾₃)
4	3	dnege-P3.30 276	1 ⊢ ((𝛾₁ ∧ 𝛾₂) → 𝛾₃)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132   ∧ 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-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by: (None)