Theorem testbitestn 823

Description: A proposition is testable iff its negation is testable. See also dcn 746 (which could be read as "Decidability implies testability"). (Contributed by David A. Wheeler, 6-Dec-2018.)

Assertion

Ref	Expression
testbitestn	⊢ (DECID ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑)

Proof of Theorem testbitestn

Step	Hyp	Ref	Expression
1		notnotnot 628	. . . 4 ⊢ (¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)
2	1	orbi2i 679	. . 3 ⊢ ((¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑) ↔ (¬ ¬ 𝜑 ∨ ¬ 𝜑))
3		orcom 647	. . 3 ⊢ ((¬ ¬ 𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4	2, 3	bitri 173	. 2 ⊢ ((¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
5		df-dc 743	. 2 ⊢ (DECID ¬ ¬ 𝜑 ↔ (¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑))
6		df-dc 743	. 2 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
7	4, 5, 6	3bitr4ri 202	1 ⊢ (DECID ¬ 𝜑 ↔ DECID ¬ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)