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Theorem cnvin 5270
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 5269 . . 3  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  `' ( A  \  B ) )
2 cnvdif 5269 . . . 4  |-  `' ( A  \  B )  =  ( `' A  \  `' B )
32difeq2i 3454 . . 3  |-  ( `' A  \  `' ( A  \  B ) )  =  ( `' A  \  ( `' A  \  `' B
) )
41, 3eqtri 2455 . 2  |-  `' ( A  \  ( A 
\  B ) )  =  ( `' A  \  ( `' A  \  `' B ) )
5 dfin4 3573 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65cnveqi 5038 . 2  |-  `' ( A  i^i  B )  =  `' ( A 
\  ( A  \  B ) )
7 dfin4 3573 . 2  |-  ( `' A  i^i  `' B
)  =  ( `' A  \  ( `' A  \  `' B
) )
84, 6, 73eqtr4i 2465 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3309    i^i cin 3311   `'ccnv 4868
This theorem is referenced by:  rnin  5272  dminxp  5302  imainrect  5303  cnvcnv  5314  pjdm  16922  ordtrest2  17256  ustexsym  18233  trust  18247  elrn3  25375  pprodcnveq  25678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877
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