Description: Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009) (Revised by Mario Carneiro, 12-Oct-2016) (Proof shortened by Wolf Lammen, 3-May-2018)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sbft | |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | spsbe | |- ( [ y / x ] ph -> E. x ph ) |
|
| 2 | 19.9t | |- ( F/ x ph -> ( E. x ph <-> ph ) ) |
|
| 3 | 1 2 | imbitrid | |- ( F/ x ph -> ( [ y / x ] ph -> ph ) ) |
| 4 | nf5r | |- ( F/ x ph -> ( ph -> A. x ph ) ) |
|
| 5 | stdpc4 | |- ( A. x ph -> [ y / x ] ph ) |
|
| 6 | 4 5 | syl6 | |- ( F/ x ph -> ( ph -> [ y / x ] ph ) ) |
| 7 | 3 6 | impbid | |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) ) |