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|Description: This is our first
definition, which introduces and defines the
biconditional connective ↔. We define a wff of the
(φ ↔ ψ) as an abbreviation for
¬ ((φ → ψ) → ¬ (ψ → φ)).
Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as." Instead, we will later use the biconditional connective for this purpose (df-or 789 is its first use), as it allows us to use logic to manipulate definitions directly. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions. Of course, we cannot use this mechanism to define the biconditional itself, since it hasn't been introduced yet. Instead, we use a more general form of definition, described as follows.
In its most general form, a definition is simply an assertion that introduces a new symbol (or a new combination of existing symbols, as in df-3an 905) that is eliminable and does not strengthen the existing language. The latter requirement means that the set of provable statements not containing the new symbol (or new combination) should remain exactly the same after the definition is introduced. Our definition of the biconditional may look unusual compared to most definitions, but it strictly satisfies these requirements.
The justification for our definition is that if we mechanically replace (φ ↔ ψ) (the definiendum i.e. the thing being defined) with ¬ ((φ → ψ) → ¬ (ψ → φ)) (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem bijust 549. It is impossible to use df-bi 109 to prove any statement expressed in the original language that can't be proved from the original axioms, because if we simply replace each instance of df-bi 109 in the proof with the corresponding bijust 549 instance, we will end up with a proof from the original axioms.
Note that from Metamath's point of view, a definition is just another axiom - i.e. an assertion we claim to be true - but from our high level point of view, we are are not strengthening the language. To indicate this fact, we prefix definition labels with "df-" instead of "ax-". (This prefixing is an informal convention that means nothing to the Metamath proof verifier; it is just for human readability.)
See dfbi1 778, dfbi2 366, and dfbi3 825 for theorems suggesting typical textbook definitions of ↔, showing that our definition has the properties we expect. Theorem dfbi 367 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.
|df-bi||⊢ (((φ ↔ ψ) → ((φ → ψ) ∧ (ψ → φ))) ∧ (((φ → ψ) ∧ (ψ → φ)) → (φ ↔ ψ)))|
|1||wph||. . . 4 wff φ|
|2||wps||. . . 4 wff ψ|
|3||1, 2||wb 97||. . 3 wff (φ ↔ ψ)|
|4||1, 2||wi 4||. . . 4 wff (φ → ψ)|
|5||2, 1||wi 4||. . . 4 wff (ψ → φ)|
|6||4, 5||wa 96||. . 3 wff ((φ → ψ) ∧ (ψ → φ))|
|7||3, 6||wi 4||. 2 wff ((φ ↔ ψ) → ((φ → ψ) ∧ (ψ → φ)))|
|8||6, 3||wi 4||. 2 wff (((φ → ψ) ∧ (ψ → φ)) → (φ ↔ ψ))|
|9||7, 8||wa 96||1 wff (((φ ↔ ψ) → ((φ → ψ) ∧ (ψ → φ))) ∧ (((φ → ψ) ∧ (ψ → φ)) → (φ ↔ ψ)))|
|Colors of variables: wff set class|
|This definition is referenced by: bi1 110 bi3 111 bi2 120 dfbi2 366|
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