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Theorem reusv2lem5 5102
Description: Lemma for reusv2 5103. (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 1661 . . . . . . . . 9
2 biimt 352 . . . . . . . . 9 ((𝐶𝐴 ∧ ⊤) → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
31, 2mpan2 682 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
4 ibar 524 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ (𝐶𝐴𝑥 = 𝐶)))
53, 4bitr3d 273 . . . . . . 7 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶)))
6 eleq1 2894 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
76pm5.32ri 571 . . . . . . 7 ((𝑥𝐴𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶))
85, 7syl6bbr 281 . . . . . 6 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
98ralimi 3161 . . . . 5 (∀𝑦𝐵 𝐶𝐴 → ∀𝑦𝐵 (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
10 ralbi 3278 . . . . 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 2659 . . 3 (∀𝑦𝐵 𝐶𝐴 → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
13 r19.28zv 4288 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ (𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1413eubidv 2659 . . 3 (𝐵 ≠ ∅ → (∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1512, 14sylan9bb 505 . 2 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
161biantrur 526 . . . . 5 (𝑥 = 𝐶 ↔ (⊤ ∧ 𝑥 = 𝐶))
1716rexbii 3251 . . . 4 (∃𝑦𝐵 𝑥 = 𝐶 ↔ ∃𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
1817reubii 3340 . . 3 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
19 reusv2lem4 5101 . . 3 (∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
2018, 19bitri 267 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
21 df-reu 3124 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶))
2215, 20, 213bitr4g 306 1 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wtru 1657  wcel 2164  ∃!weu 2639  wne 2999  wral 3117  wrex 3118  ∃!wreu 3119  c0 4144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5013  ax-pow 5065
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-nul 4145
This theorem is referenced by:  reusv2  5103
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