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 )