Theorem elcnvcnvintab

Description: Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020)

	Ref	Expression
Assertion	elcnvcnvintab	|- ( A e. `' `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. x ) ) )

Proof

Step	Hyp	Ref	Expression
1		cnvcnv	|- `' `' |^| { x | ph } = ( |^| { x | ph } i^i ( _V X. _V ) )
2		incom	|- ( |^| { x | ph } i^i ( _V X. _V ) ) = ( ( _V X. _V ) i^i |^| { x | ph } )
3	1 2	eqtri	|- `' `' |^| { x | ph } = ( ( _V X. _V ) i^i |^| { x | ph } )
4	3	eleq2i	|- ( A e. `' `' |^| { x | ph } <-> A e. ( ( _V X. _V ) i^i |^| { x | ph } ) )
5		elinintab	|- ( A e. ( ( _V X. _V ) i^i |^| { x | ph } ) <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. x ) ) )
6	4 5	bitri	|- ( A e. `' `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. x ) ) )