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Theorem dfin2 4009
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4008. Another version is given by dfin4 4016. (Contributed by NM, 10-Jun-2004.)
Assertion
Ref Expression
dfin2 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))

Proof of Theorem dfin2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3354 . . . . . 6 𝑥 ∈ V
2 eldif 3733 . . . . . 6 (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥𝐵))
31, 2mpbiran 688 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
43con2bii 346 . . . 4 (𝑥𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵))
54anbi2i 609 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
6 eldif 3733 . . 3 (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
75, 6bitr4i 267 . 2 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
87ineqri 3957 1 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  cin 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-in 3730
This theorem is referenced by:  dfun3  4014  dfin3  4015  invdif  4017  difundi  4028  difindi  4030  difdif2  4033
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