Metamath Proof Explorer


Theorem ereldm

Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996) (Revised by Mario Carneiro, 12-Aug-2015)

Ref Expression
Hypotheses ereldm.1 φ R Er X
ereldm.2 φ A R = B R
Assertion ereldm φ A X B X

Proof

Step Hyp Ref Expression
1 ereldm.1 φ R Er X
2 ereldm.2 φ A R = B R
3 2 neeq1d φ A R B R
4 ecdmn0 A dom R A R
5 ecdmn0 B dom R B R
6 3 4 5 3bitr4g φ A dom R B dom R
7 erdm R Er X dom R = X
8 1 7 syl φ dom R = X
9 8 eleq2d φ A dom R A X
10 8 eleq2d φ B dom R B X
11 6 9 10 3bitr3d φ A X B X