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Theorem reusv2 5280
Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦) that is constant for those 𝑦𝐵 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 3288 . . . 4 𝑦{𝑦𝐵𝜑}
2 nfcv 2900 . . . 4 𝑧{𝑦𝐵𝜑}
3 nfv 1921 . . . 4 𝑧 𝐶𝐴
4 nfcsb1v 3824 . . . . 5 𝑦𝑧 / 𝑦𝐶
54nfel1 2916 . . . 4 𝑦𝑧 / 𝑦𝐶𝐴
6 csbeq1a 3814 . . . . 5 (𝑦 = 𝑧𝐶 = 𝑧 / 𝑦𝐶)
76eleq1d 2818 . . . 4 (𝑦 = 𝑧 → (𝐶𝐴𝑧 / 𝑦𝐶𝐴))
81, 2, 3, 5, 7cbvralfw 3336 . . 3 (∀𝑦 ∈ {𝑦𝐵𝜑}𝐶𝐴 ↔ ∀𝑧 ∈ {𝑦𝐵𝜑}𝑧 / 𝑦𝐶𝐴)
9 rabid 3282 . . . . . 6 (𝑦 ∈ {𝑦𝐵𝜑} ↔ (𝑦𝐵𝜑))
109imbi1i 353 . . . . 5 ((𝑦 ∈ {𝑦𝐵𝜑} → 𝐶𝐴) ↔ ((𝑦𝐵𝜑) → 𝐶𝐴))
11 impexp 454 . . . . 5 (((𝑦𝐵𝜑) → 𝐶𝐴) ↔ (𝑦𝐵 → (𝜑𝐶𝐴)))
1210, 11bitri 278 . . . 4 ((𝑦 ∈ {𝑦𝐵𝜑} → 𝐶𝐴) ↔ (𝑦𝐵 → (𝜑𝐶𝐴)))
1312ralbii2 3079 . . 3 (∀𝑦 ∈ {𝑦𝐵𝜑}𝐶𝐴 ↔ ∀𝑦𝐵 (𝜑𝐶𝐴))
148, 13bitr3i 280 . 2 (∀𝑧 ∈ {𝑦𝐵𝜑}𝑧 / 𝑦𝐶𝐴 ↔ ∀𝑦𝐵 (𝜑𝐶𝐴))
15 rabn0 4284 . 2 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∃𝑦𝐵 𝜑)
16 reusv2lem5 5279 . . 3 ((∀𝑧 ∈ {𝑦𝐵𝜑}𝑧 / 𝑦𝐶𝐴 ∧ {𝑦𝐵𝜑} ≠ ∅) → (∃!𝑥𝐴𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝐴𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶))
17 nfv 1921 . . . . . 6 𝑧 𝑥 = 𝐶
184nfeq2 2917 . . . . . 6 𝑦 𝑥 = 𝑧 / 𝑦𝐶
196eqeq2d 2750 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝑧 / 𝑦𝐶))
201, 2, 17, 18, 19cbvrexfw 3338 . . . . 5 (∃𝑦 ∈ {𝑦𝐵𝜑}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶)
219anbi1i 627 . . . . . . 7 ((𝑦 ∈ {𝑦𝐵𝜑} ∧ 𝑥 = 𝐶) ↔ ((𝑦𝐵𝜑) ∧ 𝑥 = 𝐶))
22 anass 472 . . . . . . 7 (((𝑦𝐵𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑦𝐵 ∧ (𝜑𝑥 = 𝐶)))
2321, 22bitri 278 . . . . . 6 ((𝑦 ∈ {𝑦𝐵𝜑} ∧ 𝑥 = 𝐶) ↔ (𝑦𝐵 ∧ (𝜑𝑥 = 𝐶)))
2423rexbii2 3160 . . . . 5 (∃𝑦 ∈ {𝑦𝐵𝜑}𝑥 = 𝐶 ↔ ∃𝑦𝐵 (𝜑𝑥 = 𝐶))
2520, 24bitr3i 280 . . . 4 (∃𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃𝑦𝐵 (𝜑𝑥 = 𝐶))
2625reubii 3295 . . 3 (∃!𝑥𝐴𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
271, 2, 17, 18, 19cbvralfw 3336 . . . . 5 (∀𝑦 ∈ {𝑦𝐵𝜑}𝑥 = 𝐶 ↔ ∀𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶)
289imbi1i 353 . . . . . . 7 ((𝑦 ∈ {𝑦𝐵𝜑} → 𝑥 = 𝐶) ↔ ((𝑦𝐵𝜑) → 𝑥 = 𝐶))
29 impexp 454 . . . . . . 7 (((𝑦𝐵𝜑) → 𝑥 = 𝐶) ↔ (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
3028, 29bitri 278 . . . . . 6 ((𝑦 ∈ {𝑦𝐵𝜑} → 𝑥 = 𝐶) ↔ (𝑦𝐵 → (𝜑𝑥 = 𝐶)))
3130ralbii2 3079 . . . . 5 (∀𝑦 ∈ {𝑦𝐵𝜑}𝑥 = 𝐶 ↔ ∀𝑦𝐵 (𝜑𝑥 = 𝐶))
3227, 31bitr3i 280 . . . 4 (∀𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∀𝑦𝐵 (𝜑𝑥 = 𝐶))
3332reubii 3295 . . 3 (∃!𝑥𝐴𝑧 ∈ {𝑦𝐵𝜑}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
3416, 26, 333bitr3g 316 . 2 ((∀𝑧 ∈ {𝑦𝐵𝜑}𝑧 / 𝑦𝐶𝐴 ∧ {𝑦𝐵𝜑} ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
3514, 15, 34syl2anbr 602 1 ((∀𝑦𝐵 (𝜑𝐶𝐴) ∧ ∃𝑦𝐵 𝜑) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wne 2935  wral 3054  wrex 3055  ∃!wreu 3056  {crab 3058  csb 3800  c0 4221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5184  ax-pow 5242
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-nul 4222
This theorem is referenced by:  cdleme25dN  38026
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