Metamath Proof Explorer


Theorem eliind2

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 )

Proof

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 )