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Theorem cnvin 6164
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 6163 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴(𝐴𝐵))
2 cnvdif 6163 . . . 4 (𝐴𝐵) = (𝐴𝐵)
32difeq2i 4123 . . 3 (𝐴(𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
41, 3eqtri 2765 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
5 dfin4 4278 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65cnveqi 5885 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
7 dfin4 4278 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
84, 6, 73eqtr4i 2775 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cdif 3948  cin 3950  ccnv 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  rnin  6166  dminxp  6200  imainrect  6201  cnvcnv  6212  cnvrescnv  6215  pjdm  21727  ordtrest2  23212  ustexsym  24224  trust  24238  ordtcnvNEW  33919  ordtrest2NEW  33922  msrf  35547  elrn3  35762  pprodcnveq  35884  tposrescnv  48779
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