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Theorem iunxsngf2 38255
 Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
iunxsngf2.1 𝑥𝐶
iunxsngf2.2 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iunxsngf2 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem iunxsngf2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4460 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 nfcv 2751 . . . . 5 𝑥𝑦
3 iunxsngf2.1 . . . . 5 𝑥𝐶
42, 3nfel 2763 . . . 4 𝑥 𝑦𝐶
5 iunxsngf2.2 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
65eleq2d 2673 . . . 4 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
74, 6rexsngf 38245 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
81, 7syl5bb 271 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
98eqrdv 2608 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∃wrex 2897  {csn 4125  ∪ 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-3an 1033  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-sbc 3403  df-sn 4126  df-iun 4457 This theorem is referenced by:  iunxsnf  38258
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