Theorem cnvin 6156

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 6155	. . 3 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵))
2		cnvdif 6155	. . . 4 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)
3	2	difeq2i 4118	. . 3 ⊢ (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
4	1, 3	eqtri 2754	. 2 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
5		dfin4 4269	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	cnveqi 5881	. 2 ⊢ ◡(𝐴 ∩ 𝐵) = ◡(𝐴 ∖ (𝐴 ∖ 𝐵))
7		dfin4 4269	. 2 ⊢ (◡𝐴 ∩ ◡𝐵) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵))
8	4, 6, 7	3eqtr4i 2764	1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

Syntax hints:   = wceq 1534   ∖ cdif 3944   ∩ cin 3946  ^◡ccnv 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690

This theorem is referenced by:  rnin  6158  dminxp  6191  imainrect  6192  cnvcnv  6203  cnvrescnv  6206  pjdm  21705  ordtrest2  23199  ustexsym  24211  trust  24225  ordtcnvNEW  33735  ordtrest2NEW  33738  msrf  35370  elrn3  35584  pprodcnveq  35707