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Theorem reusv2lem2 5403
Description: Lemma for reusv2 5407. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
Assertion
Ref Expression
reusv2lem2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reusv2lem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eunex 5394 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵)
2 exnal 1822 . . . . 5 (∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
31, 2sylib 217 . . . 4 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
4 rzal 4513 . . . . 5 (𝐴 = ∅ → ∀𝑦𝐴 𝑥 = 𝐵)
54alrimiv 1923 . . . 4 (𝐴 = ∅ → ∀𝑥𝑦𝐴 𝑥 = 𝐵)
63, 5nsyl3 138 . . 3 (𝐴 = ∅ → ¬ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
76pm2.21d 121 . 2 (𝐴 = ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
8 simpr 483 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
9 nfra1 3272 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑧 = 𝐵
10 nfra1 3272 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑥 = 𝐵
11 simpr 483 . . . . . . . . . . . . . 14 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
12 rspa 3236 . . . . . . . . . . . . . . 15 ((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) → 𝑧 = 𝐵)
1312adantr 479 . . . . . . . . . . . . . 14 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑧 = 𝐵)
1411, 13eqtr4d 2769 . . . . . . . . . . . . 13 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝑧)
15 eqeq1 2730 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1615ralbidv 3168 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
1716biimprcd 249 . . . . . . . . . . . . . 14 (∀𝑦𝐴 𝑧 = 𝐵 → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
1817ad2antrr 724 . . . . . . . . . . . . 13 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
1914, 18mpd 15 . . . . . . . . . . . 12 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → ∀𝑦𝐴 𝑥 = 𝐵)
2019exp31 418 . . . . . . . . . . 11 (∀𝑦𝐴 𝑧 = 𝐵 → (𝑦𝐴 → (𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵)))
219, 10, 20rexlimd 3254 . . . . . . . . . 10 (∀𝑦𝐴 𝑧 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221adantl 480 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
23 r19.2z 4499 . . . . . . . . . . 11 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2423ex 411 . . . . . . . . . 10 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2524adantr 479 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2622, 25impbid 211 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
2726eubidv 2575 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
2827ex 411 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
2928exlimdv 1929 . . . . 5 (𝐴 ≠ ∅ → (∃𝑧𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
30 euex 2566 . . . . . 6 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
3116cbvexvw 2033 . . . . . 6 (∃𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧𝑦𝐴 𝑧 = 𝐵)
3230, 31sylib 217 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑧𝑦𝐴 𝑧 = 𝐵)
3329, 32impel 504 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
348, 33mpbird 256 . . 3 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3534ex 411 . 2 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
367, 35pm2.61ine 3015 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1532   = wceq 1534  wex 1774  wcel 2099  ∃!weu 2557  wne 2930  wral 3051  wrex 3060  c0 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-nul 5311  ax-pow 5369
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-ne 2931  df-ral 3052  df-rex 3061  df-dif 3950  df-nul 4326
This theorem is referenced by:  reusv2lem3  5404
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