|Metamath Proof Explorer||
|Mirrors > Home > MPE Home > Th. List > df-bi||Structured version Unicode version|
|Description: Define the biconditional
The definition df-bi 178 in this section is our first definition, which introduces and defines the biconditional connective . We define a wff of the form as an abbreviation for .
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 360 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 938) 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 176. It is impossible to use df-bi 178 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 178 in the proof with the corresponding bijust 176 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 1328) and the constant false (df-fal 1329), we will be able to prove these truth table values: (trubitru 1359), (trubifal 1360), (falbitru 1361), and (falbifal 1362).
See dfbi1 185, dfbi2 610, and dfbi3 864 for theorems suggesting typical textbook definitions of , showing that our definition has the properties we expect. Theorem dfbi1 185 is particularly useful if we want to eliminate from an expression to convert it to primitives. Theorem dfbi 611 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.
Contrast with (df-or 360), (wi 4), (df-nan 1297), and (df-xor 1314) . In some sense returns true if two truth values are equal; (df-cleq 2428) returns true if two classes are equal. (Contributed by NM, 5-Aug-1993.)
|1||wph||. . . . 5|
|2||wps||. . . . 5|
|3||1, 2||wb 177||. . . 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 179 bi3 180 dfbi1 185 dfbi1gb 186|
|Copyright terms: Public domain||W3C validator|