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Theorem unieq 3639
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 = 𝐵 𝐴 = 𝐵)

Proof of Theorem unieq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2558 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝑥 ↔ ∃𝑥𝐵 𝑦𝑥))
21abbidv 2202 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝑥})
3 dfuni2 3632 . 2 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥}
4 dfuni2 3632 . 2 𝐵 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝑥}
52, 3, 43eqtr4g 2142 1 (𝐴 = 𝐵 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  {cab 2071  wrex 2356   cuni 3630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-uni 3631
This theorem is referenced by:  unieqi  3640  unieqd  3641  uniintsnr  3701  iununir  3788  treq  3910  limeq  4171  uniex  4231  uniexg  4232  ordsucunielexmid  4313  onsucuni2  4346  elvvuni  4463  unielrel  4915  unixp0im  4924  iotass  4954  nnsucuniel  6191  en1bg  6450  bj-uniex  11165  bj-uniexg  11166  nnpredcl  11246  nnsf  11251  peano3nninf  11253  exmidsbthr  11269
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