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Theorem uneqri 3079
Description: Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
uneqri.1 ((x A x B) ↔ x 𝐶)
Assertion
Ref Expression
uneqri (AB) = 𝐶
Distinct variable groups:   x,A   x,B   x,𝐶

Proof of Theorem uneqri
StepHypRef Expression
1 elun 3078 . . 3 (x (AB) ↔ (x A x B))
2 uneqri.1 . . 3 ((x A x B) ↔ x 𝐶)
31, 2bitri 173 . 2 (x (AB) ↔ x 𝐶)
43eqriv 2034 1 (AB) = 𝐶
Colors of variables: wff set class
Syntax hints:  wb 98   wo 628   = wceq 1242   wcel 1390  cun 2909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916
This theorem is referenced by:  unidm  3080  uncom  3081  unass  3094  undi  3179  unab  3198  un0  3245
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