Theorem cnvin 6048

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 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

Proof of Theorem cnvin

Step	Hyp	Ref	Expression
1		cnvdif 6047	. . 3 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵))
2		cnvdif 6047	. . . 4 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)
3	2	difeq2i 4054	. . 3 ⊢ (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
4	1, 3	eqtri 2766	. 2 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
5		dfin4 4201	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	cnveqi 5783	. 2 ⊢ ◡(𝐴 ∩ 𝐵) = ◡(𝐴 ∖ (𝐴 ∖ 𝐵))
7		dfin4 4201	. 2 ⊢ (◡𝐴 ∩ ◡𝐵) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
8	4, 6, 7	3eqtr4i 2776	1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

Syntax hints:   = wceq 1539   ∖ cdif 3884   ∩ cin 3886  ^◡ccnv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597

This theorem is referenced by:  rnin  6050  dminxp  6083  imainrect  6084  cnvcnv  6095  cnvrescnv  6098  pjdm  20914  ordtrest2  22355  ustexsym  23367  trust  23381  ordtcnvNEW  31870  ordtrest2NEW  31873  msrf  33504  elrn3  33729  pprodcnveq  34185