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Theorem cnvin 5323
Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
cnvin  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )

Proof of Theorem cnvin
StepHypRef Expression
1 cnvdif 5322 . . 3  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  `' ( A  \  B ) )
2 cnvdif 5322 . . . 4  |-  `' ( A  \  B )  =  ( `' A  \  `' B )
32difeq2i 3533 . . 3  |-  ( `' A  \  `' ( A  \  B ) )  =  ( `' A  \  ( `' A  \  `' B
) )
41, 3eqtri 2411 . 2  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  ( `' A  \  `' B ) )
5 dfin4 3663 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65cnveqi 5090 . 2  |-  `' ( A  i^i  B )  =  `' ( A 
\  ( A  \  B ) )
7 dfin4 3663 . 2  |-  ( `' A  i^i  `' B
)  =  ( `' A  \  ( `' A  \  `' B
) )
84, 6, 73eqtr4i 2421 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    \ cdif 3386    i^i cin 3388   `'ccnv 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921
This theorem is referenced by:  rnin  5325  dminxp  5357  imainrect  5358  cnvcnv  5368  pjdm  18829  ordtrest2  19791  ustexsym  20803  trust  20817  ordtcnvNEW  28056  ordtrest2NEW  28059  msrf  29091  elrn3  29358  pprodcnveq  29686
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