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Theorem iuneq1i 38286
 Description: Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
iuneq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
iuneq1i 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iuneq1i
StepHypRef Expression
1 iuneq1i.1 . 2 𝐴 = 𝐵
2 iuneq1 4470 . 2 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
31, 2ax-mp 5 1 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-iun 4457 This theorem is referenced by:  ovolval4lem1  39539
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