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Theorem reusv2lem5 5293
Description: Lemma for reusv2 5294. (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 1535 . . . . . . . . 9
2 biimt 363 . . . . . . . . 9 ((𝐶𝐴 ∧ ⊤) → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
31, 2mpan2 689 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶)))
4 ibar 531 . . . . . . . 8 (𝐶𝐴 → (𝑥 = 𝐶 ↔ (𝐶𝐴𝑥 = 𝐶)))
53, 4bitr3d 283 . . . . . . 7 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶)))
6 eleq1 2898 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
76pm5.32ri 578 . . . . . . 7 ((𝑥𝐴𝑥 = 𝐶) ↔ (𝐶𝐴𝑥 = 𝐶))
85, 7syl6bbr 291 . . . . . 6 (𝐶𝐴 → (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
98ralimi 3158 . . . . 5 (∀𝑦𝐵 𝐶𝐴 → ∀𝑦𝐵 (((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ (𝑥𝐴𝑥 = 𝐶)))
10 ralbi 3165 . . . . 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 2666 . . 3 (∀𝑦𝐵 𝐶𝐴 → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶)))
13 r19.28zv 4444 . . . 4 (𝐵 ≠ ∅ → (∀𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ (𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1413eubidv 2666 . . 3 (𝐵 ≠ ∅ → (∃!𝑥𝑦𝐵 (𝑥𝐴𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
1512, 14sylan9bb 512 . 2 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶)))
161biantrur 533 . . . . 5 (𝑥 = 𝐶 ↔ (⊤ ∧ 𝑥 = 𝐶))
1716rexbii 3245 . . . 4 (∃𝑦𝐵 𝑥 = 𝐶 ↔ ∃𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
1817reubii 3390 . . 3 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶))
19 reusv2lem4 5292 . . 3 (∃!𝑥𝐴𝑦𝐵 (⊤ ∧ 𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
2018, 19bitri 277 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴 ∧ ⊤) → 𝑥 = 𝐶))
21 df-reu 3143 . 2 (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝑥 = 𝐶))
2215, 20, 213bitr4g 316 1 ((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1531  wtru 1532  wcel 2108  ∃!weu 2647  wne 3014  wral 3136  wrex 3137  ∃!wreu 3138  c0 4289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-nul 5201  ax-pow 5257
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-nul 4290
This theorem is referenced by:  reusv2  5294
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