Theorem bj-notbi 10045

Description: Equivalence property for negation. TODO: minimize all theorems using notbid 592 and notbii 594. (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 121	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2	1	con3d 561	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 → ¬ 𝜓))
3		bi1 111	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	con3d 561	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
5	2, 4	impbid 120	1 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98

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

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bj-notbii  10046  bj-notbid  10047  bj-dcbi  10048