Theorem opreu2reu1

Description: Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023)

	Ref	Expression
Hypothesis	opreu2reu1.a	|- ( p = <. x , y >. -> ( ch <-> ph ) )
Assertion	opreu2reu1	|- ( E! x e. A , y e. B ph <-> E! p e. ( A X. B ) ch )

Proof

Step	Hyp	Ref	Expression
1		opreu2reu1.a	|- ( p = <. x , y >. -> ( ch <-> ph ) )
2		df-2reu	|- ( E! x e. A , y e. B ph <-> ( E! x e. A E. y e. B ph /\ E! y e. B E. x e. A ph ) )
3	1	opreu2reurex	|- ( E! p e. ( A X. B ) ch <-> ( E! x e. A E. y e. B ph /\ E! y e. B E. x e. A ph ) )
4	2 3	bitr4i	|- ( E! x e. A , y e. B ph <-> E! p e. ( A X. B ) ch )