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Theorem reusv2lem1 5404
Description: Lemma for reusv2 5409. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reusv2lem1 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reusv2lem1
StepHypRef Expression
1 n0 4359 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
2 nfra1 3282 . . . . 5 𝑦𝑦𝐴 𝑥 = 𝐵
32nfmov 2558 . . . 4 𝑦∃*𝑥𝑦𝐴 𝑥 = 𝐵
4 rsp 3245 . . . . . . 7 (∀𝑦𝐴 𝑥 = 𝐵 → (𝑦𝐴𝑥 = 𝐵))
54com12 32 . . . . . 6 (𝑦𝐴 → (∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
65alrimiv 1925 . . . . 5 (𝑦𝐴 → ∀𝑥(∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
7 mo2icl 3723 . . . . 5 (∀𝑥(∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵) → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
86, 7syl 17 . . . 4 (𝑦𝐴 → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
93, 8exlimi 2215 . . 3 (∃𝑦 𝑦𝐴 → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
101, 9sylbi 217 . 2 (𝐴 ≠ ∅ → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
11 df-eu 2567 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝑦𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝑦𝐴 𝑥 = 𝐵))
1211rbaib 538 . 2 (∃*𝑥𝑦𝐴 𝑥 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))
1310, 12syl 17 1 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wex 1776  wcel 2106  ∃*wmo 2536  ∃!weu 2566  wne 2938  wral 3059  c0 4339
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-v 3480  df-dif 3966  df-nul 4340
This theorem is referenced by: (None)
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