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Theorem iineq1 3983
 Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iineq1 (A = Bx A C = x B C)
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   C(x)

Proof of Theorem iineq1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 raleq 2807 . . 3 (A = B → (x A y Cx B y C))
21abbidv 2467 . 2 (A = B → {y x A y C} = {y x B y C})
3 df-iin 3972 . 2 x A C = {y x A y C}
4 df-iin 3972 . 2 x B C = {y x B y C}
52, 3, 43eqtr4g 2410 1 (A = Bx A C = x B C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∩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-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-iin 3972 This theorem is referenced by:  iinrab2  4029  riin0  4039
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