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Theorem eusvobj2 6542
 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 4204 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧})
2 eleq2 2677 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧}))
3 abid 2598 . . . . . 6 (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ ∃𝑦𝐴 𝑥 = 𝐵)
4 velsn 4141 . . . . . 6 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
52, 3, 43bitr3g 301 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝑧))
6 nfre1 2988 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
76nfab 2755 . . . . . . . 8 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵}
87nfeq1 2764 . . . . . . 7 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧}
9 eusvobj1.1 . . . . . . . . 9 𝐵 ∈ V
109elabrex 6405 . . . . . . . 8 (𝑦𝐴𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵})
11 eleq2 2677 . . . . . . . . 9 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧}))
129elsn 4140 . . . . . . . . . 10 (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧)
13 eqcom 2617 . . . . . . . . . 10 (𝐵 = 𝑧𝑧 = 𝐵)
1412, 13bitri 263 . . . . . . . . 9 (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵)
1511, 14syl6bb 275 . . . . . . . 8 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵))
1610, 15syl5ib 233 . . . . . . 7 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦𝐴𝑧 = 𝐵))
178, 16ralrimi 2940 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦𝐴 𝑧 = 𝐵)
18 eqeq1 2614 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1918ralbidv 2969 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
2017, 19syl5ibrcom 236 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
215, 20sylbid 229 . . . 4 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221exlimiv 1845 . . 3 (∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
231, 22sylbi 206 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
24 euex 2482 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
25 rexn0 4026 . . . 4 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
2625exlimiv 1845 . . 3 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
27 r19.2z 4012 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2827ex 449 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2924, 26, 283syl 18 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
3023, 29impbid 201 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875  df-sn 4126 This theorem is referenced by:  eusvobj1  6543
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