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Theorem cnvin 5087
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 5086 . . 3  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  `' ( A  \  B ) )
2 cnvdif 5086 . . . 4  |-  `' ( A  \  B )  =  ( `' A  \  `' B )
32difeq2i 3292 . . 3  |-  ( `' A  \  `' ( A  \  B ) )  =  ( `' A  \  ( `' A  \  `' B
) )
41, 3eqtri 2304 . 2  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  ( `' A  \  `' B ) )
5 dfin4 3410 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65cnveqi 4855 . 2  |-  `' ( A  i^i  B )  =  `' ( A 
\  ( A  \  B ) )
7 dfin4 3410 . 2  |-  ( `' A  i^i  `' B
)  =  ( `' A  \  ( `' A  \  `' B
) )
84, 6, 73eqtr4i 2314 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    = wceq 1624    \ cdif 3150    i^i cin 3152   `'ccnv 4687
This theorem is referenced by:  rnin  5089  dminxp  5117  imainrect  5118  cnvcnv  5125  pjdm  16601  ordtrest2  16928  pprodcnveq  23831
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696
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