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Theorem elcnvcnvintab 36259
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
StepHypRef Expression
1 cnvcnv 5295 . . . 4  |-  `' `' |^| { x  |  ph }  =  ( |^| { x  |  ph }  i^i  ( _V  X.  _V ) )
2 incom 3616 . . . 4  |-  ( |^| { x  |  ph }  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  |^| { x  | 
ph } )
31, 2eqtri 2493 . . 3  |-  `' `' |^| { x  |  ph }  =  ( ( _V  X.  _V )  i^i  |^| { x  |  ph } )
43eleq2i 2541 . 2  |-  ( A  e.  `' `' |^| { x  |  ph }  <->  A  e.  ( ( _V 
X.  _V )  i^i  |^| { x  |  ph }
) )
5 elinintab 36252 . 2  |-  ( A  e.  ( ( _V 
X.  _V )  i^i  |^| { x  |  ph }
)  <->  ( A  e.  ( _V  X.  _V )  /\  A. x (
ph  ->  A  e.  x
) ) )
64, 5bitri 257 1  |-  ( A  e.  `' `' |^| { x  |  ph }  <->  ( A  e.  ( _V 
X.  _V )  /\  A. x ( ph  ->  A  e.  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    e. wcel 1904   {cab 2457   _Vcvv 3031    i^i cin 3389   |^|cint 4226    X. cxp 4837   `'ccnv 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-int 4227  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847
This theorem is referenced by:  cnvcnvintabd  36277
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