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Theorem reusv2lem2OLD 5019
Description: Obsolete proof of reusv2lem2 5018 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 5008 . . . . 5 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵)
2 exnal 1903 . . . . 5 (∃𝑥 ¬ ∀𝑦𝐴 𝑥 = 𝐵 ↔ ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
31, 2sylib 208 . . . 4 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ¬ ∀𝑥𝑦𝐴 𝑥 = 𝐵)
4 rzal 4217 . . . . 5 (𝐴 = ∅ → ∀𝑦𝐴 𝑥 = 𝐵)
54alrimiv 2004 . . . 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 479 . . . 4 ((𝐴 ≠ ∅ ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
9 euex 2631 . . . . . . 7 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
10 eqeq1 2764 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1110ralbidv 3124 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
1211cbvexv 2420 . . . . . . 7 (∃𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧𝑦𝐴 𝑧 = 𝐵)
139, 12sylib 208 . . . . . 6 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑧𝑦𝐴 𝑧 = 𝐵)
14 nfv 1992 . . . . . . . . . . . 12 𝑦 𝐴 ≠ ∅
15 nfra1 3079 . . . . . . . . . . . 12 𝑦𝑦𝐴 𝑧 = 𝐵
1614, 15nfan 1977 . . . . . . . . . . 11 𝑦(𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵)
17 nfra1 3079 . . . . . . . . . . 11 𝑦𝑦𝐴 𝑥 = 𝐵
18 simprr 813 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑥 = 𝐵)
19 rspa 3068 . . . . . . . . . . . . . . 15 ((∀𝑦𝐴 𝑧 = 𝐵𝑦𝐴) → 𝑧 = 𝐵)
2019ad2ant2lr 801 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑧 = 𝐵)
2118, 20eqtr4d 2797 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) ∧ (𝑦𝐴𝑥 = 𝐵)) → 𝑥 = 𝑧)
22 simplr 809 . . . . . . . . . . . . . 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 632 . . . . . . . . . . 11 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (𝑦𝐴 → (𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵)))
2616, 17, 25rexlimd 3164 . . . . . . . . . 10 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑧 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
27 r19.2z 4204 . . . . . . . . . . . 12 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2827ex 449 . . . . . . . . . . 11 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2928adantr 472 . . . . . . . . . 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 2627 . . . . . . . 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 2010 . . . . . 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 3015 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  ∃!weu 2607  wne 2932  wral 3050  wrex 3051  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-nul 4059
This theorem is referenced by: (None)
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