Description: Equality property for substitution, from Tarski's system. Used in proof of Theorem 9.7 in Megill p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993) Revise df-sb . (Revised by BJ, 30-Dec-2020)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sbequ | |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equequ2 | |- ( x = y -> ( u = x <-> u = y ) ) |
|
| 2 | 1 | imbi1d | |- ( x = y -> ( ( u = x -> A. z ( z = u -> ph ) ) <-> ( u = y -> A. z ( z = u -> ph ) ) ) ) |
| 3 | 2 | albidv | |- ( x = y -> ( A. u ( u = x -> A. z ( z = u -> ph ) ) <-> A. u ( u = y -> A. z ( z = u -> ph ) ) ) ) |
| 4 | df-sb | |- ( [ x / z ] ph <-> A. u ( u = x -> A. z ( z = u -> ph ) ) ) |
|
| 5 | df-sb | |- ( [ y / z ] ph <-> A. u ( u = y -> A. z ( z = u -> ph ) ) ) |
|
| 6 | 3 4 5 | 3bitr4g | |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) |