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Theorem eusvobj2 6835
Description: Specify the same property in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
eusvobj1.1 𝐵 ∈ V
Assertion
Ref Expression
eusvobj2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem eusvobj2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 euabsn2 4415 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧})
2 eleq2 2833 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧}))
3 abid 2753 . . . . . 6 (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ ∃𝑦𝐴 𝑥 = 𝐵)
4 velsn 4350 . . . . . 6 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
52, 3, 43bitr3g 304 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝑧))
6 nfre1 3151 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
76nfab 2912 . . . . . . . 8 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵}
87nfeq1 2921 . . . . . . 7 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧}
9 eusvobj1.1 . . . . . . . . 9 𝐵 ∈ V
109elabrex 6693 . . . . . . . 8 (𝑦𝐴𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵})
11 eleq2 2833 . . . . . . . . 9 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧}))
129elsn 4349 . . . . . . . . . 10 (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧)
13 eqcom 2772 . . . . . . . . . 10 (𝐵 = 𝑧𝑧 = 𝐵)
1412, 13bitri 266 . . . . . . . . 9 (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵)
1511, 14syl6bb 278 . . . . . . . 8 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵))
1610, 15syl5ib 235 . . . . . . 7 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦𝐴𝑧 = 𝐵))
178, 16ralrimi 3104 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦𝐴 𝑧 = 𝐵)
18 eqeq1 2769 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1918ralbidv 3133 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
2017, 19syl5ibrcom 238 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
215, 20sylbid 231 . . . 4 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221exlimiv 2025 . . 3 (∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
231, 22sylbi 208 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
24 euex 2591 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
25 rexn0 4233 . . . 4 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
2625exlimiv 2025 . . 3 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
27 r19.2z 4219 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2827ex 401 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2924, 26, 283syl 18 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
3023, 29impbid 203 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wex 1874  wcel 2155  ∃!weu 2581  {cab 2751  wne 2937  wral 3055  wrex 3056  Vcvv 3350  c0 4079  {csn 4334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-nul 4080  df-sn 4335
This theorem is referenced by:  eusvobj1  6836
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