Metamath Proof Explorer

Theorem inecmo2

Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018) (Revised by Peter Mazsa, 2-Sep-2021)

		Ref	Expression
	Assertion	inecmo2	\|- ( ( A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u e. A u R x /\ Rel R ) )

Proof

Step	Hyp	Ref	Expression
1		id	\|- ( u = v -> u = v )
2	1	inecmo	\|- ( Rel R -> ( A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) <-> A. x E* u e. A u R x ) )
3	2	pm5.32ri	\|- ( ( A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u e. A u R x /\ Rel R ) )

Step

Hyp

Ref

Expression

 |-  ( u = v -> u = v )

inecmo

 |-  ( Rel R -> ( A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) <-> A. x E* u e. A u R x ) )

pm5.32ri

 |-  ( ( A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u e. A u R x /\ Rel R ) )