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Theorem unieq 3900
 Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
unieq (A = BA = B)

Proof of Theorem unieq
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2808 . . 3 (A = B → (x A y xx B y x))
21abbidv 2467 . 2 (A = B → {y x A y x} = {y x B y x})
3 dfuni2 3893 . 2 A = {y x A y x}
4 dfuni2 3893 . 2 B = {y x B y x}
52, 3, 43eqtr4g 2410 1 (A = BA = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  ∪cuni 3891 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-rex 2620  df-uni 3892 This theorem is referenced by:  unieqi  3901  unieqd  3902  uniintsn  3963  iununi  4050  pw1equn  4331  pw1eqadj  4332  nnadjoin  4520  pw1fnval  5851  pw1fnf1o  5855  brtcfn  6246
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