Theorem bibif 335

Description: Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)

Assertion

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

Proof of Theorem bibif

Step	Hyp	Ref	Expression
1		nbn2 334	. 2 ⊢ (¬ ψ → (¬ φ ↔ (ψ ↔ φ)))
2		bicom 191	. 2 ⊢ ((ψ ↔ φ) ↔ (φ ↔ ψ))
3	1, 2	syl6rbb 253	1 ⊢ (¬ ψ → ((φ ↔ ψ) ↔ ¬ φ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  nbn  336