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Theorem eusvobj2 7423
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 4730 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧})
2 eleq2 2828 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧}))
3 abid 2716 . . . . . 6 (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ ∃𝑦𝐴 𝑥 = 𝐵)
4 velsn 4647 . . . . . 6 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
52, 3, 43bitr3g 313 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝑧))
6 nfre1 3283 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
76nfab 2909 . . . . . . . 8 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵}
87nfeq1 2919 . . . . . . 7 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧}
9 eusvobj1.1 . . . . . . . . 9 𝐵 ∈ V
109elabrex 7262 . . . . . . . 8 (𝑦𝐴𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵})
11 eleq2 2828 . . . . . . . . 9 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧}))
129elsn 4646 . . . . . . . . . 10 (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧)
13 eqcom 2742 . . . . . . . . . 10 (𝐵 = 𝑧𝑧 = 𝐵)
1412, 13bitri 275 . . . . . . . . 9 (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵)
1511, 14bitrdi 287 . . . . . . . 8 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵))
1610, 15imbitrid 244 . . . . . . 7 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦𝐴𝑧 = 𝐵))
178, 16ralrimi 3255 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦𝐴 𝑧 = 𝐵)
18 eqeq1 2739 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1918ralbidv 3176 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
2017, 19syl5ibrcom 247 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
215, 20sylbid 240 . . . 4 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221exlimiv 1928 . . 3 (∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
231, 22sylbi 217 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
24 euex 2575 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
25 rexn0 4517 . . . 4 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
2625exlimiv 1928 . . 3 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
27 r19.2z 4501 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2827ex 412 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2924, 26, 283syl 18 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
3023, 29impbid 212 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wex 1776  wcel 2106  ∃!weu 2566  {cab 2712  wne 2938  wral 3059  wrex 3068  Vcvv 3478  c0 4339  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340  df-sn 4632
This theorem is referenced by:  eusvobj1  7424
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