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Theorem difin2 4220
Description: Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
difin2 (𝐴𝐶 → (𝐴𝐵) = ((𝐶𝐵) ∩ 𝐴))

Proof of Theorem difin2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3907 . . . . 5 (𝐴𝐶 → (𝑥𝐴𝑥𝐶))
21pm4.71d 565 . . . 4 (𝐴𝐶 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐶)))
32anbi1d 633 . . 3 (𝐴𝐶 → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵)))
4 eldif 3890 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 elin 3896 . . . 4 (𝑥 ∈ ((𝐶𝐵) ∩ 𝐴) ↔ (𝑥 ∈ (𝐶𝐵) ∧ 𝑥𝐴))
6 eldif 3890 . . . . 5 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶 ∧ ¬ 𝑥𝐵))
76anbi1i 627 . . . 4 ((𝑥 ∈ (𝐶𝐵) ∧ 𝑥𝐴) ↔ ((𝑥𝐶 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐴))
8 ancom 464 . . . . 5 (((𝑥𝐶 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐴) ↔ (𝑥𝐴 ∧ (𝑥𝐶 ∧ ¬ 𝑥𝐵)))
9 anass 472 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ↔ (𝑥𝐴 ∧ (𝑥𝐶 ∧ ¬ 𝑥𝐵)))
108, 9bitr4i 281 . . . 4 (((𝑥𝐶 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐴) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
115, 7, 103bitri 300 . . 3 (𝑥 ∈ ((𝐶𝐵) ∩ 𝐴) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
123, 4, 113bitr4g 317 . 2 (𝐴𝐶 → (𝑥 ∈ (𝐴𝐵) ↔ 𝑥 ∈ ((𝐶𝐵) ∩ 𝐴)))
1312eqrdv 2736 1 (𝐴𝐶 → (𝐴𝐵) = ((𝐶𝐵) ∩ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wcel 2111  cdif 3877  cin 3879  wss 3880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3422  df-dif 3883  df-in 3887  df-ss 3897
This theorem is referenced by:  gsumdifsnd  19370  issubdrg  19849  restcld  22093  limcnlp  24799  symgcom2  31096  difelsiga  31837  sigapildsyslem  31865  ldgenpisyslem1  31867  difelcarsg2  32016  ballotlemfp1  32194  asindmre  35623  caragendifcl  43755  gsumdifsndf  45076
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