Description: This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" -. A. x x = y as an antecedent. ph normally has z free and can be read ph ( z ) , and ps substitutes y for z and can be read ph ( y ) . We do not require that x and y be distinct: if they are not, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.
To obtain a closed-theorem form of this inference, prefix the hypotheses with A. x A. z , conjoin them, and apply dvelimdf .
Other variants of this theorem are dvelimh (with no distinct variable restrictions) and dvelimhw (that avoids ax-13 ). Usage of this theorem is discouraged because it depends on ax-13 . Check out dvelimhw for a version requiring fewer axioms. (Contributed by NM, 23-Nov-1994) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | dvelim.1 | |- ( ph -> A. x ph ) |
|
| dvelim.2 | |- ( z = y -> ( ph <-> ps ) ) |
||
| Assertion | dvelim | |- ( -. A. x x = y -> ( ps -> A. x ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dvelim.1 | |- ( ph -> A. x ph ) |
|
| 2 | dvelim.2 | |- ( z = y -> ( ph <-> ps ) ) |
|
| 3 | ax-5 | |- ( ps -> A. z ps ) |
|
| 4 | 1 3 2 | dvelimh | |- ( -. A. x x = y -> ( ps -> A. x ps ) ) |