Theorem rcp-RAA2 516

Description: Reductio ad Absurdum.

Hypotheses

Ref	Expression
rcp-RAA2.1	⊢ ((𝛾₁ ∧ ¬ 𝛾₂) → 𝜑)
rcp-RAA2.2	⊢ ((𝛾₁ ∧ ¬ 𝛾₂) → ¬ 𝜑)

Assertion

Ref	Expression
rcp-RAA2	⊢ (𝛾₁ → 𝛾₂)

Proof of Theorem rcp-RAA2

Step	Hyp	Ref	Expression
1		rcp-RAA2.1	. . 3 ⊢ ((𝛾₁ ∧ ¬ 𝛾₂) → 𝜑)
2		rcp-RAA2.2	. . 3 ⊢ ((𝛾₁ ∧ ¬ 𝛾₂) → ¬ 𝜑)
3	1, 2	rcp-NDNEGI2 219	. 2 ⊢ (𝛾₁ → ¬ ¬ 𝛾₂)
4	3	dnege-P3.30 276	1 ⊢ (𝛾₁ → 𝛾₂)

Syntax hints:  ¬ wff-neg 9   → 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-or-D2.3 145  df-true-D2.4 155

This theorem is referenced by: (None)