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Theorem reusv2lem3 5192
Description: Lemma for reusv2 5195. (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 485 . . . 4 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
2 nfv 1892 . . . . . 6 𝑥𝑦𝐴 𝐵 ∈ V
3 nfeu1 2635 . . . . . 6 𝑥∃!𝑥𝑦𝐴 𝑥 = 𝐵
42, 3nfan 1881 . . . . 5 𝑥(∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
5 euex 2622 . . . . . . . 8 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
6 rexn0 4368 . . . . . . . . 9 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
76exlimiv 1908 . . . . . . . 8 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
8 r19.2z 4354 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
98ex 413 . . . . . . . 8 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
105, 7, 93syl 18 . . . . . . 7 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
1110adantl 482 . . . . . 6 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
12 nfra1 3186 . . . . . . . 8 𝑦𝑦𝐴 𝐵 ∈ V
13 nfre1 3269 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
1413nfeu 2639 . . . . . . . 8 𝑦∃!𝑥𝑦𝐴 𝑥 = 𝐵
1512, 14nfan 1881 . . . . . . 7 𝑦(∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
16 rsp 3172 . . . . . . . . . . . . . 14 (∀𝑦𝐴 𝐵 ∈ V → (𝑦𝐴𝐵 ∈ V))
1716impcom 408 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ ∀𝑦𝐴 𝐵 ∈ V) → 𝐵 ∈ V)
18 isset 3449 . . . . . . . . . . . . 13 (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
1917, 18sylib 219 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ ∀𝑦𝐴 𝐵 ∈ V) → ∃𝑥 𝑥 = 𝐵)
2019adantrr 713 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → ∃𝑥 𝑥 = 𝐵)
21 rspe 3267 . . . . . . . . . . . . . . 15 ((𝑦𝐴𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2221ex 413 . . . . . . . . . . . . . 14 (𝑦𝐴 → (𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2322ancrd 552 . . . . . . . . . . . . 13 (𝑦𝐴 → (𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
2423eximdv 1895 . . . . . . . . . . . 12 (𝑦𝐴 → (∃𝑥 𝑥 = 𝐵 → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)))
2524imp 407 . . . . . . . . . . 11 ((𝑦𝐴 ∧ ∃𝑥 𝑥 = 𝐵) → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
2620, 25syldan 591 . . . . . . . . . 10 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
27 eupick 2688 . . . . . . . . . 10 ((∃!𝑥𝑦𝐴 𝑥 = 𝐵 ∧ ∃𝑥(∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵)) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
281, 26, 27syl2an2 682 . . . . . . . . 9 ((𝑦𝐴 ∧ (∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)) → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
2928ex 413 . . . . . . . 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 3185 . . . . . 6 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
3211, 31impbid 213 . . . . 5 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑦𝐴 𝑥 = 𝐵))
334, 32eubid 2633 . . . 4 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
341, 33mpbird 258 . . 3 ((∀𝑦𝐴 𝐵 ∈ V ∧ ∃!𝑥𝑦𝐴 𝑥 = 𝐵) → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3534ex 413 . 2 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
36 reusv2lem2 5191 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)
3735, 36impbid1 226 1 (∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wex 1761  wcel 2081  ∃!weu 2611  wne 2984  wral 3105  wrex 3106  Vcvv 3437  c0 4211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-nul 5101  ax-pow 5157
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-v 3439  df-dif 3862  df-nul 4212
This theorem is referenced by:  reusv2lem4  5193  eusv4  5198
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