Theorem bii 140

Description: Relate the biconditional connective to primitive connectives. See biigb 129 for an unusual version proved directly from axioms.

Assertion

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

Proof of Theorem bii

Step	Hyp	Ref	Expression
1		bi1 130	. . 3 ⊢ ((φ ↔ ψ) → (φ → ψ))
2		bi2 131	. . 3 ⊢ ((φ ↔ ψ) → (ψ → φ))
3	1, 2	jc 119	. 2 ⊢ ((φ ↔ ψ) → ¬ ((φ → ψ) → ¬ (ψ → φ)))
4		bi3 132	. . 3 ⊢ ((φ → ψ) → ((ψ → φ) → (φ ↔ ψ)))
5	4	impi 124	. 2 ⊢ (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ))
6	3, 5	impbi 139	1 ⊢ ((φ ↔ ψ) ↔ ¬ ((φ → ψ) → ¬ (ψ → φ)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127

This theorem is referenced by:  bi 396

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

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