Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | eliind2.1 | |- F/ x ph |
|
eliind2.2 | |- ( ph -> A e. V ) |
||
eliind2.3 | |- ( ( ph /\ x e. B ) -> A e. C ) |
||
Assertion | eliind2 | |- ( ph -> A e. |^|_ x e. B C ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliind2.1 | |- F/ x ph |
|
2 | eliind2.2 | |- ( ph -> A e. V ) |
|
3 | eliind2.3 | |- ( ( ph /\ x e. B ) -> A e. C ) |
|
4 | 3 | ex | |- ( ph -> ( x e. B -> A e. C ) ) |
5 | 1 4 | ralrimi | |- ( ph -> A. x e. B A e. C ) |
6 | eliin | |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) ) |
|
7 | 2 6 | syl | |- ( ph -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) ) |
8 | 5 7 | mpbird | |- ( ph -> A e. |^|_ x e. B C ) |