Description: Equivalence for substitution when y is not free in ph . The implication "to the left" is sb2 and does not require the non-freeness hypothesis. Theorem sb6 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, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | sb6f.1 | |- F/ y ph |
|
| Assertion | sb6f | |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sb6f.1 | |- F/ y ph |
|
| 2 | 1 | nf5ri | |- ( ph -> A. y ph ) |
| 3 | 2 | sbimi | |- ( [ y / x ] ph -> [ y / x ] A. y ph ) |
| 4 | sb4a | |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) ) |
|
| 5 | 3 4 | syl | |- ( [ y / x ] ph -> A. x ( x = y -> ph ) ) |
| 6 | sb2 | |- ( A. x ( x = y -> ph ) -> [ y / x ] ph ) |
|
| 7 | 5 6 | impbii | |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) |