Metamath Proof Explorer


Theorem iuneq2dv

Description: Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004)

Ref Expression
Hypothesis iuneq2dv.1
|- ( ( ph /\ x e. A ) -> B = C )
Assertion iuneq2dv
|- ( ph -> U_ x e. A B = U_ x e. A C )

Proof

Step Hyp Ref Expression
1 iuneq2dv.1
 |-  ( ( ph /\ x e. A ) -> B = C )
2 1 ralrimiva
 |-  ( ph -> A. x e. A B = C )
3 iuneq2
 |-  ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )
4 2 3 syl
 |-  ( ph -> U_ x e. A B = U_ x e. A C )