Theorem raa-P4 514

Description: Reductio ad Absurdum.

This rule combines ndnegi-P3.3 168 with double negative elimination, and is thus dependent on the Law of Excluded Middle.

Hypotheses

Ref	Expression
raa-P4.1	⊢ ((𝛾 ∧ ¬ 𝜑) → 𝜓)
raa-P4.2	⊢ ((𝛾 ∧ ¬ 𝜑) → ¬ 𝜓)

Assertion

Ref	Expression
raa-P4	⊢ (𝛾 → 𝜑)

Proof of Theorem raa-P4

Step	Hyp	Ref	Expression
1		raa-P4.1	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜑) → 𝜓)
2		raa-P4.2	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜑) → ¬ 𝜓)
3	1, 2	ndnegi-P3.3 168	. 2 ⊢ (𝛾 → ¬ ¬ 𝜑)
4	3	dnege-P3.30 276	1 ⊢ (𝛾 → 𝜑)

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)