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Theorem reusv2lem2OLD 4900
Description: Obsolete proof of reusv2lem2 4899 as of 7-Aug-2021. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
reusv2lem2OLD (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reusv2lem2OLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eunex 4889 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵)
2 exnal 1794 . . . . 5 (∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
31, 2sylib 208 . . . 4 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
4 rzal 4106 . . . . 5 (𝐴 = ∅ → ∀𝑦𝐴 𝑥 = 𝐵)
54alrimiv 1895 . . . 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 476 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
9 euex 2522 . . . . . . 7 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
10 eqeq1 2655 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1110ralbidv 3015 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
1211cbvexv 2311 . . . . . . 7 (∃𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧𝑦𝐴 𝑧 = 𝐵)
139, 12sylib 208 . . . . . 6 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑧𝑦𝐴 𝑧 = 𝐵)
14 nfv 1883 . . . . . . . . . . . 12 𝑦 𝐴 ≠ ∅
15 nfra1 2970 . . . . . . . . . . . 12 𝑦𝑦𝐴 𝑧 = 𝐵
1614, 15nfan 1868 . . . . . . . . . . 11 𝑦(𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵)
17 nfra1 2970 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑥 = 𝐵
18 simprr 811 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑥 = 𝐵)
19 rspa 2959 . . . . . . . . . . . . . . 15 ((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) → 𝑧 = 𝐵)
2019ad2ant2lr 799 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑧 = 𝐵)
2118, 20eqtr4d 2688 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑥 = 𝑧)
22 simplr 807 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → ∀𝑦𝐴 𝑧 = 𝐵)
2322, 11syl5ibrcom 237 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
2421, 23mpd 15 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → ∀𝑦𝐴 𝑥 = 𝐵)
2524exp32 630 . . . . . . . . . . 11 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (𝑦𝐴 → (𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵)))
2616, 17, 25rexlimd 3055 . . . . . . . . . 10 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
27 r19.2z 4093 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2827ex 449 . . . . . . . . . . 11 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2928adantr 480 . . . . . . . . . 10 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
3026, 29impbid 202 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
3130eubidv 2518 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
3231ex 449 . . . . . . 7 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
3332exlimdv 1901 . . . . . 6 (𝐴 ≠ ∅ → (∃𝑧𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
3413, 33syl5 34 . . . . 5 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
3534imp 444 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
368, 35mpbird 247 . . 3 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3736ex 449 . 2 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
387, 37pm2.61ine 2906 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  wne 2823  wral 2941  wrex 2942  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by: (None)
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