Theorem elcnvcnvintab 37888

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.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem elcnvcnvintab

Step	Hyp	Ref	Expression
1		cnvcnv 5586	. . . 4 |- `' `' |^| { x | ph } = ( |^| { x | ph } i^i ( _V X. _V ) )
2		incom 3805	. . . 4 |- ( |^| { x | ph } i^i ( _V X. _V ) ) = ( ( _V X. _V ) i^i |^| { x | ph } )
3	1, 2	eqtri 2644	. . 3 |- `' `' |^| { x | ph } = ( ( _V X. _V ) i^i |^| { x | ph } )
4	3	eleq2i 2693	. 2 |- ( A e. `' `' |^| { x | ph } <-> A e. ( ( _V X. _V ) i^i |^| { x | ph } ) )
5		elinintab 37881	. 2 |- ( A e. ( ( _V X. _V ) i^i |^| { x | ph } ) <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. x ) ) )
6	4, 5	bitri 264	1 |- ( A e. `' `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   _Vcvv 3200   i^i cin 3573   |^|cint 4475   X. cxp 5112   `'ccnv 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  cnvcnvintabd  37906