Theorem difeq 29355

Description: Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Assertion

Ref	Expression
difeq	|- ( ( A \ B ) = C <-> ( ( C i^i B ) = (/) /\ ( C u. B ) = ( A u. B ) ) )

Proof of Theorem difeq

Step	Hyp	Ref	Expression
1		incom 3805	. . . . 5 |- ( B i^i ( A \ B ) ) = ( ( A \ B ) i^i B )
2		disjdif 4040	. . . . 5 |- ( B i^i ( A \ B ) ) = (/)
3	1, 2	eqtr3i 2646	. . . 4 |- ( ( A \ B ) i^i B ) = (/)
4		ineq1 3807	. . . 4 |- ( ( A \ B ) = C -> ( ( A \ B ) i^i B ) = ( C i^i B ) )
5	3, 4	syl5reqr 2671	. . 3 |- ( ( A \ B ) = C -> ( C i^i B ) = (/) )
6		undif1 4043	. . . 4 |- ( ( A \ B ) u. B ) = ( A u. B )
7		uneq1 3760	. . . 4 |- ( ( A \ B ) = C -> ( ( A \ B ) u. B ) = ( C u. B ) )
8	6, 7	syl5reqr 2671	. . 3 |- ( ( A \ B ) = C -> ( C u. B ) = ( A u. B ) )
9	5, 8	jca 554	. 2 |- ( ( A \ B ) = C -> ( ( C i^i B ) = (/) /\ ( C u. B ) = ( A u. B ) ) )
10		simpl 473	. . . 4 |- ( ( ( C i^i B ) = (/) /\ ( C u. B ) = ( A u. B ) ) -> ( C i^i B ) = (/) )
11		disj3 4021	. . . . 5 |- ( ( C i^i B ) = (/) <-> C = ( C \ B ) )
12		eqcom 2629	. . . . 5 |- ( C = ( C \ B ) <-> ( C \ B ) = C )
13	11, 12	bitri 264	. . . 4 |- ( ( C i^i B ) = (/) <-> ( C \ B ) = C )
14	10, 13	sylib 208	. . 3 |- ( ( ( C i^i B ) = (/) /\ ( C u. B ) = ( A u. B ) ) -> ( C \ B ) = C )
15		difeq1 3721	. . . . . 6 |- ( ( C u. B ) = ( A u. B ) -> ( ( C u. B ) \ B ) = ( ( A u. B ) \ B ) )
16		difun2 4048	. . . . . 6 |- ( ( C u. B ) \ B ) = ( C \ B )
17		difun2 4048	. . . . . 6 |- ( ( A u. B ) \ B ) = ( A \ B )
18	15, 16, 17	3eqtr3g 2679	. . . . 5 |- ( ( C u. B ) = ( A u. B ) -> ( C \ B ) = ( A \ B ) )
19	18	eqeq1d 2624	. . . 4 |- ( ( C u. B ) = ( A u. B ) -> ( ( C \ B ) = C <-> ( A \ B ) = C ) )
20	19	adantl 482	. . 3 |- ( ( ( C i^i B ) = (/) /\ ( C u. B ) = ( A u. B ) ) -> ( ( C \ B ) = C <-> ( A \ B ) = C ) )
21	14, 20	mpbid 222	. 2 |- ( ( ( C i^i B ) = (/) /\ ( C u. B ) = ( A u. B ) ) -> ( A \ B ) = C )
22	9, 21	impbii 199	1 |- ( ( A \ B ) = C <-> ( ( C i^i B ) = (/) /\ ( C u. B ) = ( A u. B ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  difioo  29544