Description: Two ways of expressing substitution when y is not free in ph . The implication "to the left" is equs4 and does not require the nonfreeness hypothesis. Theorem sbalex replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 replaces it with a distinctor antecedent. (Contributed by NM, 25-Apr-2008) (Revised by Mario Carneiro, 4-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . Use sbalex instead. (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | equs45f.1 | |- F/ y ph |
|
| Assertion | equs45f | |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equs45f.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 | equs5a | |- ( 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 | equs4 | |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) ) |
|
| 8 | 6 7 | impbii | |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) |