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Theorem iuneq12d 5026
Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.) Remove DV conditions (Revised by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
iuneq12d.1 (𝜑𝐴 = 𝐵)
iuneq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
iuneq12d (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem iuneq12d
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 iuneq12d.1 . . . . . . 7 (𝜑𝐴 = 𝐵)
21eleq2d 2825 . . . . . 6 (𝜑 → (𝑥𝐴𝑥𝐵))
32anbi1d 631 . . . . 5 (𝜑 → ((𝑥𝐴𝑡𝐶) ↔ (𝑥𝐵𝑡𝐶)))
43rexbidv2 3173 . . . 4 (𝜑 → (∃𝑥𝐴 𝑡𝐶 ↔ ∃𝑥𝐵 𝑡𝐶))
54abbidv 2806 . . 3 (𝜑 → {𝑡 ∣ ∃𝑥𝐴 𝑡𝐶} = {𝑡 ∣ ∃𝑥𝐵 𝑡𝐶})
6 df-iun 4998 . . 3 𝑥𝐴 𝐶 = {𝑡 ∣ ∃𝑥𝐴 𝑡𝐶}
7 df-iun 4998 . . 3 𝑥𝐵 𝐶 = {𝑡 ∣ ∃𝑥𝐵 𝑡𝐶}
85, 6, 73eqtr4g 2800 . 2 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
9 iuneq12d.2 . . . 4 (𝜑𝐶 = 𝐷)
109adantr 480 . . 3 ((𝜑𝑥𝐵) → 𝐶 = 𝐷)
1110iuneq2dv 5021 . 2 (𝜑 𝑥𝐵 𝐶 = 𝑥𝐵 𝐷)
128, 11eqtrd 2775 1 (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = 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-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-iun 4998
This theorem is referenced by:  disjiunb  5138  cfsmolem  10308  cfsmo  10309  wunex2  10776  wuncval2  10785  imasval  17558  lpival  21352  cnextval  24085  cnextfval  24086  dvfval  25947  fedgmullem1  33657  irngval  33700  mblfinlem2  37645  heiborlem10  37807  iunrelexpmin1  43698  iunrelexpmin2  43702  colleq12d  44249
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