New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iundif2 GIF version

Theorem iundif2 4033
 Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4020 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
iundif2 x A (B C) = (B x A C)
Distinct variable group:   x,B
Allowed substitution hints:   A(x)   C(x)

Proof of Theorem iundif2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eldif 3221 . . . . 5 (y (B C) ↔ (y B ¬ y C))
21rexbii 2639 . . . 4 (x A y (B C) ↔ x A (y B ¬ y C))
3 r19.42v 2765 . . . 4 (x A (y B ¬ y C) ↔ (y B x A ¬ y C))
4 rexnal 2625 . . . . . 6 (x A ¬ y C ↔ ¬ x A y C)
5 vex 2862 . . . . . . 7 y V
6 eliin 3974 . . . . . . 7 (y V → (y x A Cx A y C))
75, 6ax-mp 8 . . . . . 6 (y x A Cx A y C)
84, 7xchbinxr 302 . . . . 5 (x A ¬ y C ↔ ¬ y x A C)
98anbi2i 675 . . . 4 ((y B x A ¬ y C) ↔ (y B ¬ y x A C))
102, 3, 93bitri 262 . . 3 (x A y (B C) ↔ (y B ¬ y x A C))
11 eliun 3973 . . 3 (y x A (B C) ↔ x A y (B C))
12 eldif 3221 . . 3 (y (B x A C) ↔ (y B ¬ y x A C))
1310, 11, 123bitr4i 268 . 2 (y x A (B C) ↔ y (B x A C))
1413eqriv 2350 1 x A (B C) = (B x A C)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ∖ cdif 3206  ∪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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-iun 3971  df-iin 3972 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator