Theorem dfbi 610

Description: Definition df-bi 177 rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008.)

Assertion

Ref	Expression
dfbi	⊢ (((φ ↔ ψ) → ((φ → ψ) ∧ (ψ → φ))) ∧ (((φ → ψ) ∧ (ψ → φ)) → (φ ↔ ψ)))

Proof of Theorem dfbi

Step	Hyp	Ref	Expression
1		dfbi2 609	. . 3 ⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ)))
2	1	biimpi 186	. 2 ⊢ ((φ ↔ ψ) → ((φ → ψ) ∧ (ψ → φ)))
3	1	biimpri 197	. 2 ⊢ (((φ → ψ) ∧ (ψ → φ)) → (φ ↔ ψ))
4	2, 3	pm3.2i 441	1 ⊢ (((φ ↔ ψ) → ((φ → ψ) ∧ (ψ → φ))) ∧ (((φ → ψ) ∧ (ψ → φ)) → (φ ↔ ψ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by: (None)