Metamath Proof Explorer


Theorem opabid

Description: The law of concretion. Special case of Theorem 9.5 of Quine p. 61. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker opabidw when possible. (Contributed by NM, 14-Apr-1995) (Proof shortened by Andrew Salmon, 25-Jul-2011) (New usage is discouraged.)

Ref Expression
Assertion opabid xyxy|φφ

Proof

Step Hyp Ref Expression
1 opex xyV
2 copsexg z=xyφxyz=xyφ
3 2 bicomd z=xyxyz=xyφφ
4 df-opab xy|φ=z|xyz=xyφ
5 1 3 4 elab2 xyxy|φφ