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Theorem reusv2lem3 5270
 Description: Lemma for reusv2 5273. (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 488 . . . 4 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
2 nfv 1915 . . . . . 6 𝑥𝑦𝐴 𝐵 ∈ V
3 nfeu1 2649 . . . . . 6 𝑥∃!𝑥𝑦𝐴 𝑥 = 𝐵
42, 3nfan 1900 . . . . 5 𝑥(∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
5 euex 2637 . . . . . . . 8 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
6 rexn0 4415 . . . . . . . . 9 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
76exlimiv 1931 . . . . . . . 8 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
8 r19.2z 4401 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
98ex 416 . . . . . . . 8 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
105, 7, 93syl 18 . . . . . . 7 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
1110adantl 485 . . . . . 6 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
12 nfra1 3183 . . . . . . . 8 𝑦𝑦𝐴 𝐵 ∈ V
13 nfre1 3266 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
1413nfeuw 2654 . . . . . . . 8 𝑦∃!𝑥𝑦𝐴 𝑥 = 𝐵
1512, 14nfan 1900 . . . . . . 7 𝑦(∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
16 rsp 3170 . . . . . . . . . . . . . 14 (∀𝑦𝐴 𝐵 ∈ V → (𝑦𝐴𝐵 ∈ V))
1716impcom 411 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ ∀𝑦𝐴 𝐵 ∈ V) → 𝐵 ∈ V)
18 isset 3454 . . . . . . . . . . . . 13 (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
1917, 18sylib 221 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ∀𝑦𝐴 𝐵 ∈ V) → ∃𝑥 𝑥 = 𝐵)
2019adantrr 716 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → ∃𝑥 𝑥 = 𝐵)
21 rspe 3264 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2221ex 416 . . . . . . . . . . . . . 14 (𝑦𝐴 → (𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2322ancrd 555 . . . . . . . . . . . . 13 (𝑦𝐴 → (𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
2423eximdv 1918 . . . . . . . . . . . 12 (𝑦𝐴 → (∃𝑥 𝑥 = 𝐵 → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
2524imp 410 . . . . . . . . . . 11 ((𝑦𝐴 ∧ ∃𝑥 𝑥 = 𝐵) → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
2620, 25syldan 594 . . . . . . . . . 10 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
27 eupick 2695 . . . . . . . . . 10 ((∃!𝑥𝑦𝐴 𝑥 = 𝐵 ∧ ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
281, 26, 27syl2an2 685 . . . . . . . . 9 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
2928ex 416 . . . . . . . 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 3182 . . . . . 6 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
3211, 31impbid 215 . . . . 5 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑦𝐴 𝑥 = 𝐵))
334, 32eubid 2648 . . . 4 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
341, 33mpbird 260 . . 3 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3534ex 416 . 2 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
36 reusv2lem2 5269 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3735, 36impbid1 228 1 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3442  ∅c0 4246 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5178  ax-pow 5235 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-nul 4247 This theorem is referenced by:  reusv2lem4  5271  eusv4  5276
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