Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim not depending on ax-10 , but with disjoint variables. (Contributed by Wolf Lammen, 28-Jan-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | sbrim.1 | |- F/ x ph |
|
| Assertion | sbrimv | |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbrim.1 | |- F/ x ph |
|
| 2 | 1 | 19.21 | |- ( A. x ( ph -> ( x = y -> ps ) ) <-> ( ph -> A. x ( x = y -> ps ) ) ) |
| 3 | 2 | sbrimvlem | |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) |