Theorem nbn 542

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		pm2.21 71	. . 3 ⊢ (¬ ψ → (ψ → φ))
2		nbn.1	. . . . 5 ⊢ ¬ φ
3	2	a1i 7	. . . 4 ⊢ (¬ ψ → ¬ φ)
4	3	pm2.21d 74	. . 3 ⊢ (¬ ψ → (φ → ψ))
5	1, 4	impbid 397	. 2 ⊢ (¬ ψ → (ψ ↔ φ))
6		bi1 130	. . 3 ⊢ ((ψ ↔ φ) → (ψ → φ))
7	2, 6	mtoi 94	. 2 ⊢ ((ψ ↔ φ) → ¬ ψ)
8	5, 7	impbi 139	1 ⊢ (¬ ψ ↔ (ψ ↔ φ))

Syntax hints:  ¬ wn 1   ↔ wb 127

This theorem is referenced by:  n0f 1713  disj 1733  dm0rn0 2549  reldm0 2550  intirr 2628

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

This theorem depends on definitions:  df-bi 128  df-an 198

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