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|Description: Define the biconditional
The definition df-bi 179 in this section is our first definition, which
introduces and defines the biconditional connective
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 361 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 941) 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 (the definiendum i.e. the thing being defined) with (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem bijust 177. It is impossible to use df-bi 179 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 179 in the proof with the corresponding bijust 177 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 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 a naming convention for human readability.)
After we define the constant true (df-tru 1337) and the constant false (df-fal 1338), we will be able to prove these truth table values: (trubitru 1368), (trubifal 1369), (falbitru 1370), and (falbifal 1371).
See dfbi1 186, dfbi2 611, and dfbi3 865
for theorems suggesting typical
textbook definitions of
|1||wph||. . . . 5|
|2||wps||. . . . 5|
|3||1, 2||wb 178||. . . 4|
|4||1, 2||wi 4||. . . . . 6|
|5||2, 1||wi 4||. . . . . . 7|
|6||5||wn 3||. . . . . 6|
|7||4, 6||wi 4||. . . . 5|
|8||7||wn 3||. . . 4|
|9||3, 8||wi 4||. . 3|
|10||8, 3||wi 4||. . . 4|
|11||10||wn 3||. . 3|
|12||9, 11||wi 4||. 2|
|Colors of variables: wff set class|
|This definition is referenced by: bi1 180 bi3 181 dfbi1 186 dfbi1gb 187|
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