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Theorem cnvin 5104
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 5103 . . 3  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  `' ( A  \  B ) )
2 cnvdif 5103 . . . 4  |-  `' ( A  \  B )  =  ( `' A  \  `' B )
32difeq2i 3304 . . 3  |-  ( `' A  \  `' ( A  \  B ) )  =  ( `' A  \  ( `' A  \  `' B
) )
41, 3eqtri 2316 . 2  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  ( `' A  \  `' B ) )
5 dfin4 3422 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65cnveqi 4872 . 2  |-  `' ( A  i^i  B )  =  `' ( A 
\  ( A  \  B ) )
7 dfin4 3422 . 2  |-  ( `' A  i^i  `' B
)  =  ( `' A  \  ( `' A  \  `' B
) )
84, 6, 73eqtr4i 2326 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    \ cdif 3162    i^i cin 3164   `'ccnv 4704
This theorem is referenced by:  rnin  5106  dminxp  5134  imainrect  5135  cnvcnv  5142  pjdm  16623  ordtrest2  16950  elrn3  24191  pprodcnveq  24494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713
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