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Theorem eusvobj2 7400
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 4729 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧})
2 eleq2 2822 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧}))
3 abid 2713 . . . . . 6 (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ ∃𝑦𝐴 𝑥 = 𝐵)
4 velsn 4644 . . . . . 6 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
52, 3, 43bitr3g 312 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝑧))
6 nfre1 3282 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
76nfab 2909 . . . . . . . 8 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵}
87nfeq1 2918 . . . . . . 7 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧}
9 eusvobj1.1 . . . . . . . . 9 𝐵 ∈ V
109elabrex 7241 . . . . . . . 8 (𝑦𝐴𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵})
11 eleq2 2822 . . . . . . . . 9 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧}))
129elsn 4643 . . . . . . . . . 10 (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧)
13 eqcom 2739 . . . . . . . . . 10 (𝐵 = 𝑧𝑧 = 𝐵)
1412, 13bitri 274 . . . . . . . . 9 (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵)
1511, 14bitrdi 286 . . . . . . . 8 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵))
1610, 15imbitrid 243 . . . . . . 7 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦𝐴𝑧 = 𝐵))
178, 16ralrimi 3254 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦𝐴 𝑧 = 𝐵)
18 eqeq1 2736 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1918ralbidv 3177 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
2017, 19syl5ibrcom 246 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
215, 20sylbid 239 . . . 4 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221exlimiv 1933 . . 3 (∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
231, 22sylbi 216 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
24 euex 2571 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
25 rexn0 4510 . . . 4 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
2625exlimiv 1933 . . 3 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
27 r19.2z 4494 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2827ex 413 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2924, 26, 283syl 18 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
3023, 29impbid 211 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2562  {cab 2709  wne 2940  wral 3061  wrex 3070  Vcvv 3474  c0 4322  {csn 4628
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-nul 4323  df-sn 4629
This theorem is referenced by:  eusvobj1  7401
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