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Theorem reusv2lem3 4901
Description: Lemma for reusv2 4904. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reusv2lem3 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reusv2lem3
StepHypRef Expression
1 simpr 476 . . . 4 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
2 nfv 1883 . . . . . 6 𝑥𝑦𝐴 𝐵 ∈ V
3 nfeu1 2508 . . . . . 6 𝑥∃!𝑥𝑦𝐴 𝑥 = 𝐵
42, 3nfan 1868 . . . . 5 𝑥(∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
5 euex 2522 . . . . . . . 8 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
6 rexn0 4107 . . . . . . . . 9 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
76exlimiv 1898 . . . . . . . 8 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
8 r19.2z 4093 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
98ex 449 . . . . . . . 8 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
105, 7, 93syl 18 . . . . . . 7 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
1110adantl 481 . . . . . 6 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
12 nfra1 2970 . . . . . . . 8 𝑦𝑦𝐴 𝐵 ∈ V
13 nfre1 3034 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
1413nfeu 2514 . . . . . . . 8 𝑦∃!𝑥𝑦𝐴 𝑥 = 𝐵
1512, 14nfan 1868 . . . . . . 7 𝑦(∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
16 rsp 2958 . . . . . . . . . . . . . 14 (∀𝑦𝐴 𝐵 ∈ V → (𝑦𝐴𝐵 ∈ V))
1716impcom 445 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ ∀𝑦𝐴 𝐵 ∈ V) → 𝐵 ∈ V)
18 isset 3238 . . . . . . . . . . . . 13 (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
1917, 18sylib 208 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ∀𝑦𝐴 𝐵 ∈ V) → ∃𝑥 𝑥 = 𝐵)
2019adantrr 753 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → ∃𝑥 𝑥 = 𝐵)
21 rspe 3032 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2221ex 449 . . . . . . . . . . . . . 14 (𝑦𝐴 → (𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2322ancrd 576 . . . . . . . . . . . . 13 (𝑦𝐴 → (𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
2423eximdv 1886 . . . . . . . . . . . 12 (𝑦𝐴 → (∃𝑥 𝑥 = 𝐵 → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
2524imp 444 . . . . . . . . . . 11 ((𝑦𝐴 ∧ ∃𝑥 𝑥 = 𝐵) → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
2620, 25syldan 486 . . . . . . . . . 10 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
27 eupick 2565 . . . . . . . . . 10 ((∃!𝑥𝑦𝐴 𝑥 = 𝐵 ∧ ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
281, 26, 27syl2an2 892 . . . . . . . . 9 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
2928ex 449 . . . . . . . 8 (𝑦𝐴 → ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
3029com3l 89 . . . . . . 7 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → (𝑦𝐴𝑥 = 𝐵)))
3115, 13, 30ralrimd 2988 . . . . . 6 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
3211, 31impbid 202 . . . . 5 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑦𝐴 𝑥 = 𝐵))
334, 32eubid 2516 . . . 4 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
341, 33mpbird 247 . . 3 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3534ex 449 . 2 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
36 reusv2lem2 4899 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3735, 36impbid1 215 1 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  wne 2823  wral 2941  wrex 2942  Vcvv 3231  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:  reusv2lem4  4902  eusv4  4907
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