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Theorem uneqri 3062
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 3061 . . 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 2019 1 (AB) = 𝐶
Colors of variables: wff set class
Syntax hints:  wb 98   wo 616   = wceq 1228   wcel 1374  cun 2892
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899
This theorem is referenced by:  unidm  3063  uncom  3064  unass  3077  undi  3162  unab  3181  un0  3228
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