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Theorem iinxprg 4043
 Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
iinxprg.1 (x = AC = D)
iinxprg.2 (x = BC = E)
Assertion
Ref Expression
iinxprg ((A V B W) → x {A, B}C = (DE))
Distinct variable groups:   x,A   x,B   x,D   x,E
Allowed substitution hints:   C(x)   V(x)   W(x)

Proof of Theorem iinxprg
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 iinxprg.1 . . . . 5 (x = AC = D)
21eleq2d 2420 . . . 4 (x = A → (y Cy D))
3 iinxprg.2 . . . . 5 (x = BC = E)
43eleq2d 2420 . . . 4 (x = B → (y Cy E))
52, 4ralprg 3775 . . 3 ((A V B W) → (x {A, B}y C ↔ (y D y E)))
6 vex 2862 . . . 4 y V
7 eliin 3974 . . . 4 (y V → (y x {A, B}Cx {A, B}y C))
86, 7ax-mp 5 . . 3 (y x {A, B}Cx {A, B}y C)
9 elin 3219 . . 3 (y (DE) ↔ (y D y E))
105, 8, 93bitr4g 279 . 2 ((A V B W) → (y x {A, B}Cy (DE)))
1110eqrdv 2351 1 ((A V B W) → x {A, B}C = (DE))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ∩ cin 3208  {cpr 3738  ∩ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-3an 936  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-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-sn 3741  df-pr 3742  df-iin 3972 This theorem is referenced by: (None)
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