Description: Variable substitution for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Alexander van der Vekens, 17-Jun-2017) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | sb8eu.1 | |- F/ y ph |
|
Assertion | sb8mo | |- ( E* x ph <-> E* y [ y / x ] ph ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8eu.1 | |- F/ y ph |
|
2 | 1 | sb8e | |- ( E. x ph <-> E. y [ y / x ] ph ) |
3 | 1 | sb8eu | |- ( E! x ph <-> E! y [ y / x ] ph ) |
4 | 2 3 | imbi12i | |- ( ( E. x ph -> E! x ph ) <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) ) |
5 | moeu | |- ( E* x ph <-> ( E. x ph -> E! x ph ) ) |
|
6 | moeu | |- ( E* y [ y / x ] ph <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) ) |
|
7 | 4 5 6 | 3bitr4i | |- ( E* x ph <-> E* y [ y / x ] ph ) |