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Theorem reusv2lem2 5290
Description: Lemma for reusv2 5294. (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 5281 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵)
2 exnal 1820 . . . . 5 (∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
31, 2sylib 220 . . . 4 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
4 rzal 4451 . . . . 5 (𝐴 = ∅ → ∀𝑦𝐴 𝑥 = 𝐵)
54alrimiv 1921 . . . 4 (𝐴 = ∅ → ∀𝑥𝑦𝐴 𝑥 = 𝐵)
63, 5nsyl3 140 . . 3 (𝐴 = ∅ → ¬ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
76pm2.21d 121 . 2 (𝐴 = ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
8 simpr 487 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
9 nfra1 3217 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑧 = 𝐵
10 nfra1 3217 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑥 = 𝐵
11 simpr 487 . . . . . . . . . . . . . 14 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
12 rspa 3204 . . . . . . . . . . . . . . 15 ((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) → 𝑧 = 𝐵)
1312adantr 483 . . . . . . . . . . . . . 14 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑧 = 𝐵)
1411, 13eqtr4d 2857 . . . . . . . . . . . . 13 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝑧)
15 eqeq1 2823 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1615ralbidv 3195 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
1716biimprcd 252 . . . . . . . . . . . . . 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 422 . . . . . . . . . . 11 (∀𝑦𝐴 𝑧 = 𝐵 → (𝑦𝐴 → (𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵)))
219, 10, 20rexlimd 3315 . . . . . . . . . 10 (∀𝑦𝐴 𝑧 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221adantl 484 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
23 r19.2z 4438 . . . . . . . . . . 11 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2423ex 415 . . . . . . . . . 10 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2524adantr 483 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2622, 25impbid 214 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
2726eubidv 2666 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
2827ex 415 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
2928exlimdv 1927 . . . . 5 (𝐴 ≠ ∅ → (∃𝑧𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
30 euex 2656 . . . . . 6 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
3116cbvexvw 2037 . . . . . 6 (∃𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧𝑦𝐴 𝑧 = 𝐵)
3230, 31sylib 220 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑧𝑦𝐴 𝑧 = 𝐵)
3329, 32impel 508 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
348, 33mpbird 259 . . 3 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3534ex 415 . 2 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
367, 35pm2.61ine 3098 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1528   = wceq 1530  wex 1773  wcel 2107  ∃!weu 2647  wne 3014  wral 3136  wrex 3137  c0 4289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201  ax-pow 5257
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-dif 3937  df-nul 4290
This theorem is referenced by:  reusv2lem3  5291
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