Theorem bii 158

Description: Relate the biconditional connective to primitive connectives. See biigb 159 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 148	. . 3 ⊢ ((φ ↔ ψ) → (φ → ψ))
2		bi2 149	. . 3 ⊢ ((φ ↔ ψ) → (ψ → φ))
3	1, 2	jc 138	. 2 ⊢ ((φ ↔ ψ) → ¬ ((φ → ψ) → ¬ (ψ → φ)))
4		bi3 150	. . 3 ⊢ ((φ → ψ) → ((ψ → φ) → (φ ↔ ψ)))
5	4	impi 143	. 2 ⊢ (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ))
6	3, 5	impbi 157	1 ⊢ ((φ ↔ ψ) ↔ ¬ ((φ → ψ) → ¬ (ψ → φ)))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146

This theorem is referenced by:  bi 514

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

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