Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | raleqbidva.1 | |- ( ph -> A = B ) |
|
raleqbidva.2 | |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) |
||
Assertion | rexeqbidva | |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbidva.1 | |- ( ph -> A = B ) |
|
2 | raleqbidva.2 | |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) |
|
3 | 2 | rexbidva | |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) |
4 | 1 | rexeqdv | |- ( ph -> ( E. x e. A ch <-> E. x e. B ch ) ) |
5 | 3 4 | bitrd | |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) |