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Theorem reusv2lem1 4828
Description: Lemma for reusv2 4834. (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 3907 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
2 nfra1 2936 . . . . 5 𝑦𝑦𝐴 𝑥 = 𝐵
32nfmo 2486 . . . 4 𝑦∃*𝑥𝑦𝐴 𝑥 = 𝐵
4 rsp 2924 . . . . . . 7 (∀𝑦𝐴 𝑥 = 𝐵 → (𝑦𝐴𝑥 = 𝐵))
54com12 32 . . . . . 6 (𝑦𝐴 → (∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
65alrimiv 1852 . . . . 5 (𝑦𝐴 → ∀𝑥(∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
7 mo2icl 3367 . . . . 5 (∀𝑥(∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵) → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
86, 7syl 17 . . . 4 (𝑦𝐴 → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
93, 8exlimi 2084 . . 3 (∃𝑦 𝑦𝐴 → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
101, 9sylbi 207 . 2 (𝐴 ≠ ∅ → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
11 eu5 2495 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝑦𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝑦𝐴 𝑥 = 𝐵))
1211rbaib 946 . 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 196  wal 1478   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  ∃*wmo 2470  wne 2790  wral 2907  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3558  df-nul 3892
This theorem is referenced by: (None)
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