Theorem falseie-P4.22b 445

Description: '⊥' Introduction and Elimination (closed form). †

Assertion

Ref	Expression
falseie-P4.22b	⊢ ((𝜑 → ⊥) ↔ ¬ 𝜑)

Proof of Theorem falseie-P4.22b

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ (((𝜑 → ⊥) ∧ 𝜑) → 𝜑)
2		rcp-NDASM1of2 193	. . . 4 ⊢ (((𝜑 → ⊥) ∧ 𝜑) → (𝜑 → ⊥))
3	1, 2	ndime-P3.6 171	. . 3 ⊢ (((𝜑 → ⊥) ∧ 𝜑) → ⊥)
4	3	falsenegi-P4.18 432	. 2 ⊢ ((𝜑 → ⊥) → ¬ 𝜑)
5		rcp-NDASM1of1 192	. . 3 ⊢ (¬ 𝜑 → ¬ 𝜑)
6	5	impoe-P4.4a 377	. 2 ⊢ (¬ 𝜑 → (𝜑 → ⊥))
7	4, 6	rcp-NDBII0 239	1 ⊢ ((𝜑 → ⊥) ↔ ¬ 𝜑)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104   ∧ 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-true-D2.4 155  df-false-D2.5 158

This theorem is referenced by:  peirce2exclmid-P4.41b  513