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Theorem reusv2lem5 5400
Description: Lemma for reusv2 5401. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reusv2lem5 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶
Allowed substitution hint:   𝐶(𝑦)

Proof of Theorem reusv2lem5
StepHypRef Expression
1 tru 1545 . . . . . . . . 9
2 biimt 360 . . . . . . . . 9 ((𝐶𝐴 ∧ ⊤) → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
31, 2mpan2 689 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
4 ibar 529 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ (𝐶𝐴𝑥 = 𝐶)))
53, 4bitr3d 280 . . . . . . 7 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶)))
6 eleq1 2821 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
76pm5.32ri 576 . . . . . . 7 ((𝑥𝐴𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶))
85, 7bitr4di 288 . . . . . 6 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
98ralimi 3083 . . . . 5 (∀𝑦𝐵 𝐶𝐴 → ∀𝑦𝐵 (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
10 ralbi 3103 . . . . 5 (∀𝑦𝐵 (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)) → (∀𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
119, 10syl 17 . . . 4 (∀𝑦𝐵 𝐶𝐴 → (∀𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
1211eubidv 2580 . . 3 (∀𝑦𝐵 𝐶𝐴 → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
13 r19.28zv 4500 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ (𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1413eubidv 2580 . . 3 (𝐵 ≠ ∅ → (∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1512, 14sylan9bb 510 . 2 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
161biantrur 531 . . . . 5 (𝑥 = 𝐶 ↔ (⊤ ∧ 𝑥 = 𝐶))
1716rexbii 3094 . . . 4 (∃𝑦𝐵 𝑥 = 𝐶 ↔ ∃𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
1817reubii 3385 . . 3 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
19 reusv2lem4 5399 . . 3 (∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
2018, 19bitri 274 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
21 df-reu 3377 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶))
2215, 20, 213bitr4g 313 1 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wtru 1542  wcel 2106  ∃!weu 2562  wne 2940  wral 3061  wrex 3070  ∃!wreu 3374  c0 4322
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  ax-nul 5306  ax-pow 5363
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-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-nul 4323
This theorem is referenced by:  reusv2  5401
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