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Theorem reusv2lem3 5398
Description: Lemma for reusv2 5401. (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 484 . . . 4 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
2 nfv 1916 . . . . . 6 𝑥𝑦𝐴 𝐵 ∈ V
3 nfeu1 2581 . . . . . 6 𝑥∃!𝑥𝑦𝐴 𝑥 = 𝐵
42, 3nfan 1901 . . . . 5 𝑥(∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
5 euex 2570 . . . . . . . 8 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
6 rexn0 4510 . . . . . . . . 9 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
76exlimiv 1932 . . . . . . . 8 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
8 r19.2z 4494 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
98ex 412 . . . . . . . 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 3280 . . . . . . . 8 𝑦𝑦𝐴 𝐵 ∈ V
13 nfre1 3281 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
1413nfeuw 2586 . . . . . . . 8 𝑦∃!𝑥𝑦𝐴 𝑥 = 𝐵
1512, 14nfan 1901 . . . . . . 7 𝑦(∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
16 rsp 3243 . . . . . . . . . . . . . 14 (∀𝑦𝐴 𝐵 ∈ V → (𝑦𝐴𝐵 ∈ V))
1716impcom 407 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ ∀𝑦𝐴 𝐵 ∈ V) → 𝐵 ∈ V)
18 isset 3486 . . . . . . . . . . . . 13 (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
1917, 18sylib 217 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ∀𝑦𝐴 𝐵 ∈ V) → ∃𝑥 𝑥 = 𝐵)
2019adantrr 714 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → ∃𝑥 𝑥 = 𝐵)
21 rspe 3245 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2221ex 412 . . . . . . . . . . . . . 14 (𝑦𝐴 → (𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2322ancrd 551 . . . . . . . . . . . . 13 (𝑦𝐴 → (𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
2423eximdv 1919 . . . . . . . . . . . 12 (𝑦𝐴 → (∃𝑥 𝑥 = 𝐵 → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
2524imp 406 . . . . . . . . . . 11 ((𝑦𝐴 ∧ ∃𝑥 𝑥 = 𝐵) → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
2620, 25syldan 590 . . . . . . . . . 10 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
27 eupick 2628 . . . . . . . . . 10 ((∃!𝑥𝑦𝐴 𝑥 = 𝐵 ∧ ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
281, 26, 27syl2an2 683 . . . . . . . . 9 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
2928ex 412 . . . . . . . 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 3260 . . . . . 6 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
3211, 31impbid 211 . . . . 5 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑦𝐴 𝑥 = 𝐵))
334, 32eubid 2580 . . . 4 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
341, 33mpbird 257 . . 3 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3534ex 412 . 2 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
36 reusv2lem2 5397 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3735, 36impbid1 224 1 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wex 1780  wcel 2105  ∃!weu 2561  wne 2939  wral 3060  wrex 3069  Vcvv 3473  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-nul 5306  ax-pow 5363
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3475  df-dif 3951  df-nul 4323
This theorem is referenced by:  reusv2lem4  5399  eusv4  5404
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