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Theorem iunxunpr 31728
Description: Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
Hypotheses
Ref Expression
iunxunsn.1 (𝑥 = 𝑋𝐵 = 𝐶)
iunxunpr.2 (𝑥 = 𝑌𝐵 = 𝐷)
Assertion
Ref Expression
iunxunpr ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑋   𝑥,𝐷   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iunxunpr
StepHypRef Expression
1 iunxun 5091 . 2 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 𝑥 ∈ {𝑋, 𝑌}𝐵)
2 iunxunsn.1 . . . 4 (𝑥 = 𝑋𝐵 = 𝐶)
3 iunxunpr.2 . . . 4 (𝑥 = 𝑌𝐵 = 𝐷)
42, 3iunxprg 5093 . . 3 ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶𝐷))
54uneq2d 4160 . 2 ((𝑋𝑉𝑌𝑊) → ( 𝑥𝐴 𝐵 𝑥 ∈ {𝑋, 𝑌}𝐵) = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
61, 5eqtrid 2784 1 ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cun 3943  {cpr 4625   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-v 3476  df-un 3950  df-in 3952  df-ss 3962  df-sn 4624  df-pr 4626  df-iun 4993
This theorem is referenced by: (None)
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