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Theorem difeq 29738
Description: Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
Assertion
Ref Expression
difeq ((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))

Proof of Theorem difeq
StepHypRef Expression
1 incom 3966 . . . . 5 (𝐵 ∩ (𝐴𝐵)) = ((𝐴𝐵) ∩ 𝐵)
2 disjdif 4199 . . . . 5 (𝐵 ∩ (𝐴𝐵)) = ∅
31, 2eqtr3i 2788 . . . 4 ((𝐴𝐵) ∩ 𝐵) = ∅
4 ineq1 3968 . . . 4 ((𝐴𝐵) = 𝐶 → ((𝐴𝐵) ∩ 𝐵) = (𝐶𝐵))
53, 4syl5reqr 2813 . . 3 ((𝐴𝐵) = 𝐶 → (𝐶𝐵) = ∅)
6 undif1 4202 . . . 4 ((𝐴𝐵) ∪ 𝐵) = (𝐴𝐵)
7 uneq1 3921 . . . 4 ((𝐴𝐵) = 𝐶 → ((𝐴𝐵) ∪ 𝐵) = (𝐶𝐵))
86, 7syl5reqr 2813 . . 3 ((𝐴𝐵) = 𝐶 → (𝐶𝐵) = (𝐴𝐵))
95, 8jca 507 . 2 ((𝐴𝐵) = 𝐶 → ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))
10 simpl 474 . . . 4 (((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)) → (𝐶𝐵) = ∅)
11 disj3 4181 . . . . 5 ((𝐶𝐵) = ∅ ↔ 𝐶 = (𝐶𝐵))
12 eqcom 2771 . . . . 5 (𝐶 = (𝐶𝐵) ↔ (𝐶𝐵) = 𝐶)
1311, 12bitri 266 . . . 4 ((𝐶𝐵) = ∅ ↔ (𝐶𝐵) = 𝐶)
1410, 13sylib 209 . . 3 (((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)) → (𝐶𝐵) = 𝐶)
15 difeq1 3882 . . . . . 6 ((𝐶𝐵) = (𝐴𝐵) → ((𝐶𝐵) ∖ 𝐵) = ((𝐴𝐵) ∖ 𝐵))
16 difun2 4207 . . . . . 6 ((𝐶𝐵) ∖ 𝐵) = (𝐶𝐵)
17 difun2 4207 . . . . . 6 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
1815, 16, 173eqtr3g 2821 . . . . 5 ((𝐶𝐵) = (𝐴𝐵) → (𝐶𝐵) = (𝐴𝐵))
1918eqeq1d 2766 . . . 4 ((𝐶𝐵) = (𝐴𝐵) → ((𝐶𝐵) = 𝐶 ↔ (𝐴𝐵) = 𝐶))
2019adantl 473 . . 3 (((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)) → ((𝐶𝐵) = 𝐶 ↔ (𝐴𝐵) = 𝐶))
2114, 20mpbid 223 . 2 (((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)) → (𝐴𝐵) = 𝐶)
229, 21impbii 200 1 ((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  cdif 3728  cun 3729  cin 3730  c0 4078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079
This theorem is referenced by:  difioo  29927
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