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Theorem reusv2lem4 5270
 Description: Lemma for reusv2 5272. (Contributed by NM, 13-Dec-2012.)
Assertion
Ref Expression
reusv2lem4 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 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 reusv2lem4
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-reu 3116 . 2 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝜑𝑥 = 𝐶)))
2 anass 472 . . . . . 6 (((𝑦𝐵 ∧ (𝐶𝐴𝜑)) ∧ 𝑥 = 𝐶) ↔ (𝑦𝐵 ∧ ((𝐶𝐴𝜑) ∧ 𝑥 = 𝐶)))
3 rabid 3334 . . . . . . 7 (𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} ↔ (𝑦𝐵 ∧ (𝐶𝐴𝜑)))
43anbi1i 626 . . . . . 6 ((𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} ∧ 𝑥 = 𝐶) ↔ ((𝑦𝐵 ∧ (𝐶𝐴𝜑)) ∧ 𝑥 = 𝐶))
5 anass 472 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶)))
6 eleq1 2880 . . . . . . . . . 10 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
76anbi1d 632 . . . . . . . . 9 (𝑥 = 𝐶 → ((𝑥𝐴𝜑) ↔ (𝐶𝐴𝜑)))
87pm5.32ri 579 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ 𝑥 = 𝐶) ↔ ((𝐶𝐴𝜑) ∧ 𝑥 = 𝐶))
95, 8bitr3i 280 . . . . . . 7 ((𝑥𝐴 ∧ (𝜑𝑥 = 𝐶)) ↔ ((𝐶𝐴𝜑) ∧ 𝑥 = 𝐶))
109anbi2i 625 . . . . . 6 ((𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶))) ↔ (𝑦𝐵 ∧ ((𝐶𝐴𝜑) ∧ 𝑥 = 𝐶)))
112, 4, 103bitr4ri 307 . . . . 5 ((𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶))) ↔ (𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} ∧ 𝑥 = 𝐶))
1211rexbii2 3211 . . . 4 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶)) ↔ ∃𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝐶)
13 r19.42v 3306 . . . 4 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝜑𝑥 = 𝐶)))
14 nfrab1 3340 . . . . 5 𝑦{𝑦𝐵 ∣ (𝐶𝐴𝜑)}
15 nfcv 2958 . . . . 5 𝑧{𝑦𝐵 ∣ (𝐶𝐴𝜑)}
16 nfv 1915 . . . . 5 𝑧 𝑥 = 𝐶
17 nfcsb1v 3855 . . . . . 6 𝑦𝑧 / 𝑦𝐶
1817nfeq2 2975 . . . . 5 𝑦 𝑥 = 𝑧 / 𝑦𝐶
19 csbeq1a 3845 . . . . . 6 (𝑦 = 𝑧𝐶 = 𝑧 / 𝑦𝐶)
2019eqeq2d 2812 . . . . 5 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝑧 / 𝑦𝐶))
2114, 15, 16, 18, 20cbvrexfw 3387 . . . 4 (∃𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶)
2212, 13, 213bitr3i 304 . . 3 ((𝑥𝐴 ∧ ∃𝑦𝐵 (𝜑𝑥 = 𝐶)) ↔ ∃𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶)
2322eubii 2648 . 2 (∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝜑𝑥 = 𝐶)) ↔ ∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶)
24 elex 3462 . . . . . . . 8 (𝐶𝐴𝐶 ∈ V)
2524ad2antrl 727 . . . . . . 7 ((𝑦𝐵 ∧ (𝐶𝐴𝜑)) → 𝐶 ∈ V)
263, 25sylbi 220 . . . . . 6 (𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝐶 ∈ V)
2726rgen 3119 . . . . 5 𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝐶 ∈ V
28 nfv 1915 . . . . . 6 𝑧 𝐶 ∈ V
2917nfel1 2974 . . . . . 6 𝑦𝑧 / 𝑦𝐶 ∈ V
3019eleq1d 2877 . . . . . 6 (𝑦 = 𝑧 → (𝐶 ∈ V ↔ 𝑧 / 𝑦𝐶 ∈ V))
3114, 15, 28, 29, 30cbvralfw 3385 . . . . 5 (∀𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝐶 ∈ V ↔ ∀𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑧 / 𝑦𝐶 ∈ V)
3227, 31mpbi 233 . . . 4 𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑧 / 𝑦𝐶 ∈ V
33 reusv2lem3 5269 . . . 4 (∀𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑧 / 𝑦𝐶 ∈ V → (∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶))
3432, 33ax-mp 5 . . 3 (∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶)
35 df-ral 3114 . . . . 5 (∀𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∀𝑧(𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝑧 / 𝑦𝐶))
36 nfv 1915 . . . . . 6 𝑧(𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶)
3714nfcri 2946 . . . . . . 7 𝑦 𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}
3837, 18nfim 1897 . . . . . 6 𝑦(𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝑧 / 𝑦𝐶)
39 eleq1 2880 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} ↔ 𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}))
4039, 20imbi12d 348 . . . . . 6 (𝑦 = 𝑧 → ((𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ (𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝑧 / 𝑦𝐶)))
4136, 38, 40cbvalv1 2353 . . . . 5 (∀𝑦(𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑧(𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝑧 / 𝑦𝐶))
423imbi1i 353 . . . . . . . 8 ((𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ ((𝑦𝐵 ∧ (𝐶𝐴𝜑)) → 𝑥 = 𝐶))
43 impexp 454 . . . . . . . 8 (((𝑦𝐵 ∧ (𝐶𝐴𝜑)) → 𝑥 = 𝐶) ↔ (𝑦𝐵 → ((𝐶𝐴𝜑) → 𝑥 = 𝐶)))
4442, 43bitri 278 . . . . . . 7 ((𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ (𝑦𝐵 → ((𝐶𝐴𝜑) → 𝑥 = 𝐶)))
4544albii 1821 . . . . . 6 (∀𝑦(𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑦(𝑦𝐵 → ((𝐶𝐴𝜑) → 𝑥 = 𝐶)))
46 df-ral 3114 . . . . . 6 (∀𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶) ↔ ∀𝑦(𝑦𝐵 → ((𝐶𝐴𝜑) → 𝑥 = 𝐶)))
4745, 46bitr4i 281 . . . . 5 (∀𝑦(𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
4835, 41, 473bitr2i 302 . . . 4 (∀𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∀𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
4948eubii 2648 . . 3 (∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
5034, 49bitri 278 . 2 (∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
511, 23, 503bitri 300 1 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2112  ∃!weu 2631  ∀wral 3109  ∃wrex 3110  ∃!wreu 3111  {crab 3113  Vcvv 3444  ⦋csb 3831 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177  ax-pow 5234 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-nul 4247 This theorem is referenced by:  reusv2lem5  5271
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