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Theorem iunxunpr 30341
 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 4980 . 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 4982 . . 3 ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶𝐷))
54uneq2d 4090 . 2 ((𝑋𝑉𝑌𝑊) → ( 𝑥𝐴 𝐵 𝑥 ∈ {𝑋, 𝑌}𝐵) = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
61, 5syl5eq 2845 1 ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3879  {cpr 4527  ∪ ciun 4882 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-iun 4884 This theorem is referenced by: (None)
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