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Theorem iineq2 3986
 Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iineq2 (x A B = Cx A B = x A C)

Proof of Theorem iineq2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . . 5 (B = C → (y By C))
21ralimi 2689 . . . 4 (x A B = Cx A (y By C))
3 ralbi 2750 . . . 4 (x A (y By C) → (x A y Bx A y C))
42, 3syl 15 . . 3 (x A B = C → (x A y Bx A y C))
54abbidv 2467 . 2 (x A B = C → {y x A y B} = {y x A y C})
6 df-iin 3972 . 2 x A B = {y x A y B}
7 df-iin 3972 . 2 x A C = {y x A y C}
85, 6, 73eqtr4g 2410 1 (x A B = Cx A B = x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∩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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2619  df-iin 3972 This theorem is referenced by:  iineq2i  3988  iineq2d  3989
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