Metamath Proof Explorer


Theorem iuneq2df

Description: Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses iuneq2df.1
|- F/ x ph
iuneq2df.2
|- ( ( ph /\ x e. A ) -> B = C )
Assertion iuneq2df
|- ( ph -> U_ x e. A B = U_ x e. A C )

Proof

Step Hyp Ref Expression
1 iuneq2df.1
 |-  F/ x ph
2 iuneq2df.2
 |-  ( ( ph /\ x e. A ) -> B = C )
3 2 ex
 |-  ( ph -> ( x e. A -> B = C ) )
4 1 3 ralrimi
 |-  ( ph -> A. x e. A B = C )
5 iuneq2
 |-  ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )
6 4 5 syl
 |-  ( ph -> U_ x e. A B = U_ x e. A C )