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Theorem cnvin 5459
 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 (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cnvin
StepHypRef Expression
1 cnvdif 5458 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴(𝐴𝐵))
2 cnvdif 5458 . . . 4 (𝐴𝐵) = (𝐴𝐵)
32difeq2i 3687 . . 3 (𝐴(𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
41, 3eqtri 2632 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
5 dfin4 3826 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65cnveqi 5219 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
7 dfin4 3826 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
84, 6, 73eqtr4i 2642 1 (𝐴𝐵) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∖ cdif 3537   ∩ cin 3539  ◡ccnv 5037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by:  rnin  5461  dminxp  5493  imainrect  5494  cnvcnv  5505  pjdm  19870  ordtrest2  20818  ustexsym  21829  trust  21843  ordtcnvNEW  29294  ordtrest2NEW  29297  msrf  30693  elrn3  30906  pprodcnveq  31160
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