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Theorem elxpi 4800
 Description: Membership in a cross product. Uses fewer axioms than elxp 4801. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elxpi (A (B × C) → xy(A = x, y (x B y C)))
Distinct variable groups:   x,y,A   x,B,y   x,C,y

Proof of Theorem elxpi
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . . . . . 6 (z = A → (z = x, yA = x, y))
21anbi1d 685 . . . . 5 (z = A → ((z = x, y (x B y C)) ↔ (A = x, y (x B y C))))
322exbidv 1628 . . . 4 (z = A → (xy(z = x, y (x B y C)) ↔ xy(A = x, y (x B y C))))
43elabg 2986 . . 3 (A {z xy(z = x, y (x B y C))} → (A {z xy(z = x, y (x B y C))} ↔ xy(A = x, y (x B y C))))
54ibi 232 . 2 (A {z xy(z = x, y (x B y C))} → xy(A = x, y (x B y C)))
6 df-xp 4784 . . 3 (B × C) = {x, y (x B y C)}
7 df-opab 4623 . . 3 {x, y (x B y C)} = {z xy(z = x, y (x B y C))}
86, 7eqtri 2373 . 2 (B × C) = {z xy(z = x, y (x B y C))}
95, 8eleq2s 2445 1 (A (B × C) → xy(A = x, y (x B y C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ⟨cop 4561  {copab 4622   × cxp 4770 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-opab 4623  df-xp 4784 This theorem is referenced by: (None)
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