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Theorem iuneq1i 45025
Description: Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
Hypothesis
Ref Expression
iuneq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
iuneq1i 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶

Proof of Theorem iuneq1i
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 iuneq1i.1 . . . . . 6 𝐴 = 𝐵
21eleq2i 2831 . . . . 5 (𝑥𝐴𝑥𝐵)
32anbi1i 624 . . . 4 ((𝑥𝐴𝑡𝐶) ↔ (𝑥𝐵𝑡𝐶))
43rexbii2 3088 . . 3 (∃𝑥𝐴 𝑡𝐶 ↔ ∃𝑥𝐵 𝑡𝐶)
54abbii 2807 . 2 {𝑡 ∣ ∃𝑥𝐴 𝑡𝐶} = {𝑡 ∣ ∃𝑥𝐵 𝑡𝐶}
6 df-iun 4998 . 2 𝑥𝐴 𝐶 = {𝑡 ∣ ∃𝑥𝐴 𝑡𝐶}
7 df-iun 4998 . 2 𝑥𝐵 𝐶 = {𝑡 ∣ ∃𝑥𝐵 𝑡𝐶}
85, 6, 73eqtr4i 2773 1 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  wrex 3068   ciun 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-iun 4998
This theorem is referenced by:  ovolval4lem1  46605
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