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Theorem iindif2 4035
 Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4019 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
Assertion
Ref Expression
iindif2 (Ax A (B C) = (B x A C))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   C(x)

Proof of Theorem iindif2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3645 . . . 4 (A → (x A (y B ¬ y C) ↔ (y B x A ¬ y C)))
2 eldif 3221 . . . . . 6 (y (B C) ↔ (y B ¬ y C))
32bicomi 193 . . . . 5 ((y B ¬ y C) ↔ y (B C))
43ralbii 2638 . . . 4 (x A (y B ¬ y C) ↔ x A y (B C))
5 ralnex 2624 . . . . . 6 (x A ¬ y C ↔ ¬ x A y C)
6 eliun 3973 . . . . . 6 (y x A Cx A y C)
75, 6xchbinxr 302 . . . . 5 (x A ¬ y C ↔ ¬ y x A C)
87anbi2i 675 . . . 4 ((y B x A ¬ y C) ↔ (y B ¬ y x A C))
91, 4, 83bitr3g 278 . . 3 (A → (x A y (B C) ↔ (y B ¬ y x A C)))
10 vex 2862 . . . 4 y V
11 eliin 3974 . . . 4 (y V → (y x A (B C) ↔ x A y (B C)))
1210, 11ax-mp 8 . . 3 (y x A (B C) ↔ x A y (B C))
13 eldif 3221 . . 3 (y (B x A C) ↔ (y B ¬ y x A C))
149, 12, 133bitr4g 279 . 2 (A → (y x A (B C) ↔ y (B x A C)))
1514eqrdv 2351 1 (Ax A (B C) = (B x A C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ∖ cdif 3206  ∅c0 3550  ∪ciun 3969  ∩ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iun 3971  df-iin 3972 This theorem is referenced by: (None)
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