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Mirrors > Home > NFE Home > Th. List > df-bi | Unicode version |
Description: Define the biconditional
(logical 'iff').
Definition df-bi 177 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 359 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 936) 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
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
See dfbi1 184, dfbi2 609, and dfbi3 863
for theorems suggesting typical
textbook definitions of
Contrast with |
Ref | Expression |
---|---|
df-bi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph |
. . . . 5
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2 | wps |
. . . . 5
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3 | 1, 2 | wb 176 |
. . . 4
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4 | 1, 2 | wi 4 |
. . . . . 6
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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
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13 | 12 | wn 3 |
1
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Colors of variables: wff setvar class |
This definition is referenced by: bi1 178 bi3 179 dfbi1 184 dfbi1gb 185 |
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