Theorem dfbi1 184

Description: Relate the biconditional connective to primitive connectives. See dfbi1gb 185 for an unusual version proved directly from axioms. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
dfbi1	⊢ ((φ ↔ ψ) ↔ ¬ ((φ → ψ) → ¬ (ψ → φ)))

Proof of Theorem dfbi1

Step	Hyp	Ref	Expression
1		df-bi 177	. . 3 ⊢ ¬ (((φ ↔ ψ) → ¬ ((φ → ψ) → ¬ (ψ → φ))) → ¬ (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ)))
2		simplim 143	. . 3 ⊢ (¬ (((φ ↔ ψ) → ¬ ((φ → ψ) → ¬ (ψ → φ))) → ¬ (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ))) → ((φ ↔ ψ) → ¬ ((φ → ψ) → ¬ (ψ → φ))))
3	1, 2	ax-mp 5	. 2 ⊢ ((φ ↔ ψ) → ¬ ((φ → ψ) → ¬ (ψ → φ)))
4		bi3 179	. . 3 ⊢ ((φ → ψ) → ((ψ → φ) → (φ ↔ ψ)))
5	4	impi 140	. 2 ⊢ (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ))
6	3, 5	impbii 180	1 ⊢ ((φ ↔ ψ) ↔ ¬ ((φ → ψ) → ¬ (ψ → φ)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  bi2  189  dfbi2  609  tbw-bijust  1463  rb-bijust  1514  nfbidOLD  1833