Theorem bj-notbi 10716

Description: Equivalence property for negation. TODO: minimize all theorems using notbid 624 and notbii 626. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-notbi	⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Proof of Theorem bj-notbi

Step	Hyp	Ref	Expression
1		bi2 128	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2	1	con3d 593	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 → ¬ 𝜓))
3		bi1 116	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	con3d 593	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
5	2, 4	impbid 127	1 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bj-notbii  10717  bj-notbid  10718  bj-dcbi  10719