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Theorem reusv2lem2 4834
Description: Lemma for reusv2 4839. (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 4824 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵)
2 exnal 1751 . . . . 5 (∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
31, 2sylib 208 . . . 4 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
4 rzal 4050 . . . . 5 (𝐴 = ∅ → ∀𝑦𝐴 𝑥 = 𝐵)
54alrimiv 1852 . . . 4 (𝐴 = ∅ → ∀𝑥𝑦𝐴 𝑥 = 𝐵)
63, 5nsyl3 133 . . 3 (𝐴 = ∅ → ¬ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
76pm2.21d 118 . 2 (𝐴 = ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
8 simpr 477 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
9 nfra1 2936 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑧 = 𝐵
10 nfra1 2936 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑥 = 𝐵
11 simpr 477 . . . . . . . . . . . . . 14 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
12 rspa 2925 . . . . . . . . . . . . . . 15 ((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) → 𝑧 = 𝐵)
1312adantr 481 . . . . . . . . . . . . . 14 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑧 = 𝐵)
1411, 13eqtr4d 2658 . . . . . . . . . . . . 13 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝑧)
15 eqeq1 2625 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1615ralbidv 2981 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
1716biimprcd 240 . . . . . . . . . . . . . 14 (∀𝑦𝐴 𝑧 = 𝐵 → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
1817ad2antrr 761 . . . . . . . . . . . . 13 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
1914, 18mpd 15 . . . . . . . . . . . 12 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → ∀𝑦𝐴 𝑥 = 𝐵)
2019exp31 629 . . . . . . . . . . 11 (∀𝑦𝐴 𝑧 = 𝐵 → (𝑦𝐴 → (𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵)))
219, 10, 20rexlimd 3020 . . . . . . . . . 10 (∀𝑦𝐴 𝑧 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221adantl 482 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
23 r19.2z 4037 . . . . . . . . . . 11 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2423ex 450 . . . . . . . . . 10 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2524adantr 481 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2622, 25impbid 202 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
2726eubidv 2489 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
2827ex 450 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
2928exlimdv 1858 . . . . 5 (𝐴 ≠ ∅ → (∃𝑧𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
30 euex 2493 . . . . . 6 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
3116cbvexv 2274 . . . . . 6 (∃𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧𝑦𝐴 𝑧 = 𝐵)
3230, 31sylib 208 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑧𝑦𝐴 𝑧 = 𝐵)
3329, 32impel 485 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
348, 33mpbird 247 . . 3 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3534ex 450 . 2 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
367, 35pm2.61ine 2873 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  ∃!weu 2469  wne 2790  wral 2907  wrex 2908  c0 3896
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754  ax-pow 4808
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-rex 2913  df-v 3191  df-dif 3562  df-nul 3897
This theorem is referenced by:  reusv2lem3  4836
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