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Theorem reusv2lem2OLD 4830
Description: Obsolete proof of reusv2lem2 4829 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 4819 . . . . 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 4045 . . . . 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 euex 2493 . . . . . . 7 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
10 eqeq1 2625 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1110ralbidv 2980 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
1211cbvexv 2274 . . . . . . 7 (∃𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧𝑦𝐴 𝑧 = 𝐵)
139, 12sylib 208 . . . . . 6 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑧𝑦𝐴 𝑧 = 𝐵)
14 nfv 1840 . . . . . . . . . . . 12 𝑦 𝐴 ≠ ∅
15 nfra1 2936 . . . . . . . . . . . 12 𝑦𝑦𝐴 𝑧 = 𝐵
1614, 15nfan 1825 . . . . . . . . . . 11 𝑦(𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵)
17 nfra1 2936 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑥 = 𝐵
18 simprr 795 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑥 = 𝐵)
19 rspa 2925 . . . . . . . . . . . . . . 15 ((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) → 𝑧 = 𝐵)
2019ad2ant2lr 783 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑧 = 𝐵)
2118, 20eqtr4d 2658 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑥 = 𝑧)
22 simplr 791 . . . . . . . . . . . . . 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 3019 . . . . . . . . . 10 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
27 r19.2z 4032 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2827ex 450 . . . . . . . . . . 11 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2928adantr 481 . . . . . . . . . 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 2489 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
3231ex 450 . . . . . . 7 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑧 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)))
3332exlimdv 1858 . . . . . 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 445 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
368, 35mpbird 247 . . 3 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3736ex 450 . 2 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
387, 37pm2.61ine 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 3891
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 4749  ax-pow 4803
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 3188  df-dif 3558  df-nul 3892
This theorem is referenced by: (None)
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