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Theorem cnvin 5062
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 5061 . . 3  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  `' ( A  \  B ) )
2 cnvdif 5061 . . . 4  |-  `' ( A  \  B )  =  ( `' A  \  `' B )
32difeq2i 3252 . . 3  |-  ( `' A  \  `' ( A  \  B ) )  =  ( `' A  \  ( `' A  \  `' B
) )
41, 3eqtri 2276 . 2  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  ( `' A  \  `' B ) )
5 dfin4 3370 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65cnveqi 4830 . 2  |-  `' ( A  i^i  B )  =  `' ( A 
\  ( A  \  B ) )
7 dfin4 3370 . 2  |-  ( `' A  i^i  `' B
)  =  ( `' A  \  ( `' A  \  `' B
) )
84, 6, 73eqtr4i 2286 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    \ cdif 3110    i^i cin 3112   `'ccnv 4646
This theorem is referenced by:  rnin  5064  dminxp  5092  imainrect  5093  cnvcnv  5100  pjdm  16555  ordtrest2  16882  pprodcnveq  23785
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038  df-xp 4661  df-rel 4662  df-cnv 4663
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