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Theorem reusv2 5299
 Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦) that is constant for those 𝑦 ∈ 𝐵 such that 𝜑. The first antecedent ensures that the constant value belongs to the existential uniqueness domain 𝐴, and the second ensures that 𝐶(𝑦) is evaluated for at least one 𝑦. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reusv2 ((∀𝑦𝐵 (𝜑𝐶𝐴) ∧ ∃𝑦𝐵 𝜑) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem reusv2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfrab1 3389 . . . 4 𝑦{𝑦𝐵𝜑}
2 nfcv 2981 . . . 4 𝑧{𝑦𝐵𝜑}
3 nfv 1908 . . . 4 𝑧 𝐶𝐴
4 nfcsb1v 3910 . . . . 5 𝑦𝑧 / 𝑦𝐶
54nfel1 2998 . . . 4 𝑦𝑧 / 𝑦𝐶𝐴
6 csbeq1a 3900 . . . . 5 (𝑦 = 𝑧𝐶 = 𝑧 / 𝑦𝐶)
76eleq1d 2901 . . . 4 (𝑦 = 𝑧 → (𝐶𝐴𝑧 / 𝑦𝐶𝐴))
81, 2, 3, 5, 7cbvralfw 3442 . . 3 (∀𝑦 ∈ {𝑦𝐵𝜑}𝐶𝐴 ↔ ∀𝑧 ∈ {𝑦𝐵𝜑}𝑧 / 𝑦𝐶𝐴)
9 rabid 3383 . . . . . 6 (𝑦 ∈ {𝑦𝐵𝜑} ↔ (𝑦𝐵𝜑))
109imbi1i 351 . . . . 5 ((𝑦 ∈ {𝑦𝐵𝜑} → 𝐶𝐴) ↔ ((𝑦𝐵𝜑) → 𝐶𝐴))
11 impexp 451 . . . . 5 (((𝑦𝐵𝜑) → 𝐶𝐴) ↔ (𝑦𝐵 → (𝜑𝐶𝐴)))
1210, 11bitri 276 . . . 4 ((𝑦 ∈ {𝑦𝐵𝜑} → 𝐶𝐴) ↔ (𝑦𝐵 → (𝜑𝐶𝐴)))
1312ralbii2 3167 . . 3 (∀𝑦 ∈ {𝑦𝐵𝜑}𝐶𝐴 ↔ ∀𝑦𝐵 (𝜑𝐶𝐴))
148, 13bitr3i 278 . 2 (∀𝑧 ∈ {𝑦𝐵𝜑}𝑧 / 𝑦𝐶𝐴 ↔ ∀𝑦𝐵 (𝜑𝐶𝐴))
15 rabn0 4342 . 2 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∃𝑦𝐵 𝜑)
16 reusv2lem5 5298 . . 3 ((∀𝑧 ∈ {𝑦𝐵𝜑}𝑧 / 𝑦𝐶𝐴 ∧ {𝑦𝐵𝜑} ≠ ∅) → (∃!𝑥𝐴𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝐴𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶))
17 nfv 1908 . . . . . 6 𝑧 𝑥 = 𝐶
184nfeq2 2999 . . . . . 6 𝑦 𝑥 = 𝑧 / 𝑦𝐶
196eqeq2d 2836 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝑧 / 𝑦𝐶))
201, 2, 17, 18, 19cbvrexfw 3443 . . . . 5 (∃𝑦 ∈ {𝑦𝐵𝜑}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶)
219anbi1i 623 . . . . . . 7 ((𝑦 ∈ {𝑦𝐵𝜑} ∧ 𝑥 = 𝐶) ↔ ((𝑦𝐵𝜑) ∧ 𝑥 = 𝐶))
22 anass 469 . . . . . . 7 (((𝑦𝐵𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑦𝐵 ∧ (𝜑𝑥 = 𝐶)))
2321, 22bitri 276 . . . . . 6 ((𝑦 ∈ {𝑦𝐵𝜑} ∧ 𝑥 = 𝐶) ↔ (𝑦𝐵 ∧ (𝜑𝑥 = 𝐶)))
2423rexbii2 3249 . . . . 5 (∃𝑦 ∈ {𝑦𝐵𝜑}𝑥 = 𝐶 ↔ ∃𝑦𝐵 (𝜑𝑥 = 𝐶))
2520, 24bitr3i 278 . . . 4 (∃𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃𝑦𝐵 (𝜑𝑥 = 𝐶))
2625reubii 3396 . . 3 (∃!𝑥𝐴𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
271, 2, 17, 18, 19cbvralfw 3442 . . . . 5 (∀𝑦 ∈ {𝑦𝐵𝜑}𝑥 = 𝐶 ↔ ∀𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶)
289imbi1i 351 . . . . . . 7 ((𝑦 ∈ {𝑦𝐵𝜑} → 𝑥 = 𝐶) ↔ ((𝑦𝐵𝜑) → 𝑥 = 𝐶))
29 impexp 451 . . . . . . 7 (((𝑦𝐵𝜑) → 𝑥 = 𝐶) ↔ (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
3028, 29bitri 276 . . . . . 6 ((𝑦 ∈ {𝑦𝐵𝜑} → 𝑥 = 𝐶) ↔ (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
3130ralbii2 3167 . . . . 5 (∀𝑦 ∈ {𝑦𝐵𝜑}𝑥 = 𝐶 ↔ ∀𝑦𝐵 (𝜑𝑥 = 𝐶))
3227, 31bitr3i 278 . . . 4 (∀𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∀𝑦𝐵 (𝜑𝑥 = 𝐶))
3332reubii 3396 . . 3 (∃!𝑥𝐴𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
3416, 26, 333bitr3g 314 . 2 ((∀𝑧 ∈ {𝑦𝐵𝜑}𝑧 / 𝑦𝐶𝐴 ∧ {𝑦𝐵𝜑} ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
3514, 15, 34syl2anbr 598 1 ((∀𝑦𝐵 (𝜑𝐶𝐴) ∧ ∃𝑦𝐵 𝜑) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530   ∈ wcel 2107   ≠ wne 3020  ∀wral 3142  ∃wrex 3143  ∃!wreu 3144  {crab 3146  ⦋csb 3886  ∅c0 4294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206  ax-pow 5262 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-nul 4295 This theorem is referenced by:  cdleme25dN  37362
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