Description: An equivalence related to implicit substitution. Version of equsex with a disjoint variable condition, which does not require ax-13 . See equsexvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | equsalv.nf | |- F/ x ps |
|
equsalv.1 | |- ( x = y -> ( ph <-> ps ) ) |
||
Assertion | equsexv | |- ( E. x ( x = y /\ ph ) <-> ps ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalv.nf | |- F/ x ps |
|
2 | equsalv.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
3 | 2 | pm5.32i | |- ( ( x = y /\ ph ) <-> ( x = y /\ ps ) ) |
4 | 3 | exbii | |- ( E. x ( x = y /\ ph ) <-> E. x ( x = y /\ ps ) ) |
5 | ax6ev | |- E. x x = y |
|
6 | 1 | 19.41 | |- ( E. x ( x = y /\ ps ) <-> ( E. x x = y /\ ps ) ) |
7 | 5 6 | mpbiran | |- ( E. x ( x = y /\ ps ) <-> ps ) |
8 | 4 7 | bitri | |- ( E. x ( x = y /\ ph ) <-> ps ) |