Description: Version of equs45f with a disjoint variable condition, which does not require ax-13 . Note that the version of equs5 with a disjoint variable condition is actually sbalex (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | bj-equs45fv.1 | |- F/ y ph |
|
| Assertion | bj-equs45fv | |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-equs45fv.1 | |- F/ y ph |
|
| 2 | 1 | nf5ri | |- ( ph -> A. y ph ) |
| 3 | 2 | anim2i | |- ( ( x = y /\ ph ) -> ( x = y /\ A. y ph ) ) |
| 4 | 3 | eximi | |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ A. y ph ) ) |
| 5 | equs5av | |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) ) |
|
| 6 | 4 5 | syl | |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) |
| 7 | equs4v | |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) ) |
|
| 8 | 6 7 | impbii | |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) |