Theorem falseraa-P4 520

Description: Reductio ad Absurdum Using '⊥'.

This rule combines falsenegi-P4.18 432 with double negative elimination, and is thus dependent on the Law of Excluded Middle.

Hypothesis

Ref	Expression
falseraa-P4.1	⊢ ((𝛾 ∧ ¬ 𝜑) → ⊥)

Assertion

Ref	Expression
falseraa-P4	⊢ (𝛾 → 𝜑)

Proof of Theorem falseraa-P4

Step	Hyp	Ref	Expression
1		falseraa-P4.1	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜑) → ⊥)
2	1	falsenegi-P4.18 432	. 2 ⊢ (𝛾 → ¬ ¬ 𝜑)
3	2	dnege-P3.30 276	1 ⊢ (𝛾 → 𝜑)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132  ⊥wff-false 157

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-false-D2.5 158

This theorem is referenced by: (None)