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Theorem reusv2lem2 5387
Description: Lemma for reusv2 5391. (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 5378 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵)
2 exnal 1821 . . . . 5 (∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
31, 2sylib 217 . . . 4 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
4 rzal 4500 . . . . 5 (𝐴 = ∅ → ∀𝑦𝐴 𝑥 = 𝐵)
54alrimiv 1922 . . . 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 484 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
9 nfra1 3273 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑧 = 𝐵
10 nfra1 3273 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑥 = 𝐵
11 simpr 484 . . . . . . . . . . . . . 14 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
12 rspa 3237 . . . . . . . . . . . . . . 15 ((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) → 𝑧 = 𝐵)
1312adantr 480 . . . . . . . . . . . . . 14 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑧 = 𝐵)
1411, 13eqtr4d 2767 . . . . . . . . . . . . 13 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝑧)
15 eqeq1 2728 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1615ralbidv 3169 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
1716biimprcd 249 . . . . . . . . . . . . . 14 (∀𝑦𝐴 𝑧 = 𝐵 → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
1817ad2antrr 723 . . . . . . . . . . . . 13 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
1914, 18mpd 15 . . . . . . . . . . . 12 (((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) ∧ 𝑥 = 𝐵) → ∀𝑦𝐴 𝑥 = 𝐵)
2019exp31 419 . . . . . . . . . . 11 (∀𝑦𝐴 𝑧 = 𝐵 → (𝑦𝐴 → (𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵)))
219, 10, 20rexlimd 3255 . . . . . . . . . 10 (∀𝑦𝐴 𝑧 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221adantl 481 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
23 r19.2z 4486 . . . . . . . . . . 11 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2423ex 412 . . . . . . . . . 10 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2524adantr 480 . . . . . . . . 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 2572 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
2827ex 412 . . . . . 6 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
2928exlimdv 1928 . . . . 5 (𝐴 ≠ ∅ → (∃𝑧𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
30 euex 2563 . . . . . 6 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
3116cbvexvw 2032 . . . . . 6 (∃𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧𝑦𝐴 𝑧 = 𝐵)
3230, 31sylib 217 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑧𝑦𝐴 𝑧 = 𝐵)
3329, 32impel 505 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
348, 33mpbird 257 . . 3 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3534ex 412 . 2 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
367, 35pm2.61ine 3017 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  ∃!weu 2554  wne 2932  wral 3053  wrex 3062  c0 4314
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-nul 5296  ax-pow 5353
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-ne 2933  df-ral 3054  df-rex 3063  df-dif 3943  df-nul 4315
This theorem is referenced by:  reusv2lem3  5388
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