Theorem notbi 666

Description: Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)

Assertion

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

Proof of Theorem notbi

Step	Hyp	Ref	Expression
1		biimpr 130	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2	1	con3d 631	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 → ¬ 𝜓))
3		biimp 118	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	con3d 631	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
5	2, 4	impbid 129	1 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  notbid  667  notbii  668  ifbi  3556