Theorem nbn 724

Description: The negation of a wff is equivalent to the wff's equivalence to falsehood.

Hypothesis

Ref	Expression
nbn.1	⊢ ¬ φ

Assertion

Ref	Expression
nbn	⊢ (¬ ψ ↔ (ψ ↔ φ))

Proof of Theorem nbn

Step	Hyp	Ref	Expression
1		nbn.1	. . 3 ⊢ ¬ φ
2		nbn2 723	. . 3 ⊢ (¬ φ → (¬ ψ ↔ (φ ↔ ψ)))
3	1, 2	ax-mp 7	. 2 ⊢ (¬ ψ ↔ (φ ↔ ψ))
4		bicom 522	. 2 ⊢ ((φ ↔ ψ) ↔ (ψ ↔ φ))
5	3, 4	bitr 173	1 ⊢ (¬ ψ ↔ (ψ ↔ φ))

Syntax hints:  ¬ wn 2   ↔ wb 146

This theorem is referenced by:  nbn3 725  ne0f 2291  disj 2315  axnul2 2713  dm0rn0 3336  reldm0 3337  intirr 3447

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain