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Metamath Proof Explorer

Theorem bnj956

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypothesis bnj956.1 
|- ( A = B -> A. x A = B )
Assertion bnj956 
|- ( A = B -> U_ x e. A C = U_ x e. B C )

Proof

Step Hyp Ref Expression
1 bnj956.1 
 |-  ( A = B -> A. x A = B )
2 eleq2 
 |-  ( A = B -> ( x e. A <-> x e. B ) )
3 2 anbi1d 
 |-  ( A = B -> ( ( x e. A /\ y e. C ) <-> ( x e. B /\ y e. C ) ) )
4 3 alexbii 
 |-  ( A. x A = B -> ( E. x ( x e. A /\ y e. C ) <-> E. x ( x e. B /\ y e. C ) ) )
5 df-rex 
 |-  ( E. x e. A y e. C <-> E. x ( x e. A /\ y e. C ) )
6 df-rex 
 |-  ( E. x e. B y e. C <-> E. x ( x e. B /\ y e. C ) )
7 4 5 6 3bitr4g 
 |-  ( A. x A = B -> ( E. x e. A y e. C <-> E. x e. B y e. C ) )
8 1 7 syl 
 |-  ( A = B -> ( E. x e. A y e. C <-> E. x e. B y e. C ) )
9 8 abbidv 
 |-  ( A = B -> { y | E. x e. A y e. C } = { y | E. x e. B y e. C } )
10 df-iun 
 |-  U_ x e. A C = { y | E. x e. A y e. C }
11 df-iun 
 |-  U_ x e. B C = { y | E. x e. B y e. C }
12 9 10 11 3eqtr4g 
 |-  ( A = B -> U_ x e. A C = U_ x e. B C )