Description: Equivalence for substitution when y is not free in ph . The implication "to the right" is sb1 and does not require the non-freeness hypothesis. Theorem sb5 replaces the non-freeness hypothesis with a disjoint variable condition and uses less axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | sb6f.1 | |- F/ y ph |
|
| Assertion | sb5f | |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sb6f.1 | |- F/ y ph |
|
| 2 | 1 | sb6f | |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) |
| 3 | 1 | equs45f | |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) |
| 4 | 2 3 | bitr4i | |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) |