Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017) (Revised by Gino Giotto, 10-Jan-2024) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvaldvaw.1 | |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) |
|
| Assertion | cbvaldvaw | |- ( ph -> ( A. x ps <-> A. y ch ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvaldvaw.1 | |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) |
|
| 2 | 1 | ancoms | |- ( ( x = y /\ ph ) -> ( ps <-> ch ) ) |
| 3 | 2 | pm5.74da | |- ( x = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) ) |
| 4 | 3 | cbvalvw | |- ( A. x ( ph -> ps ) <-> A. y ( ph -> ch ) ) |
| 5 | 19.21v | |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) |
|
| 6 | 19.21v | |- ( A. y ( ph -> ch ) <-> ( ph -> A. y ch ) ) |
|
| 7 | 4 5 6 | 3bitr3i | |- ( ( ph -> A. x ps ) <-> ( ph -> A. y ch ) ) |
| 8 | 7 | pm5.74ri | |- ( ph -> ( A. x ps <-> A. y ch ) ) |