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Theorem iineq2d 3989
 Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
Hypotheses
Ref Expression
iineq2d.1 xφ
iineq2d.2 ((φ x A) → B = C)
Assertion
Ref Expression
iineq2d (φx A B = x A C)

Proof of Theorem iineq2d
StepHypRef Expression
1 iineq2d.1 . . 3 xφ
2 iineq2d.2 . . . 4 ((φ x A) → B = C)
32ex 423 . . 3 (φ → (x AB = C))
41, 3ralrimi 2695 . 2 (φx A B = C)
5 iineq2 3986 . 2 (x A B = Cx A B = x A C)
64, 5syl 15 1 (φx A B = x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀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:  iineq2dv  3991
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