Theorem bi3 179

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.)

Assertion

Ref	Expression
bi3	⊢ ((φ → ψ) → ((ψ → φ) → (φ ↔ ψ)))

Proof of Theorem bi3

Step	Hyp	Ref	Expression
1		df-bi 177	. . 3 ⊢ ¬ (((φ ↔ ψ) → ¬ ((φ → ψ) → ¬ (ψ → φ))) → ¬ (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ)))
2		simprim 142	. . 3 ⊢ (¬ (((φ ↔ ψ) → ¬ ((φ → ψ) → ¬ (ψ → φ))) → ¬ (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ))) → (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ)))
3	1, 2	ax-mp 5	. 2 ⊢ (¬ ((φ → ψ) → ¬ (ψ → φ)) → (φ ↔ ψ))
4	3	expi 141	1 ⊢ ((φ → ψ) → ((ψ → φ) → (φ ↔ ψ)))

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:  impbii  180  impbidd  181  dfbi1  184  bisym  281