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Theorem reusv2lem4 4832
Description: Lemma for reusv2 4834. (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 2914 . 2 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝜑𝑥 = 𝐶)))
2 anass 680 . . . . . 6 (((𝑦𝐵 ∧ (𝐶𝐴𝜑)) ∧ 𝑥 = 𝐶) ↔ (𝑦𝐵 ∧ ((𝐶𝐴𝜑) ∧ 𝑥 = 𝐶)))
3 rabid 3106 . . . . . . 7 (𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} ↔ (𝑦𝐵 ∧ (𝐶𝐴𝜑)))
43anbi1i 730 . . . . . 6 ((𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} ∧ 𝑥 = 𝐶) ↔ ((𝑦𝐵 ∧ (𝐶𝐴𝜑)) ∧ 𝑥 = 𝐶))
5 anass 680 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ 𝑥 = 𝐶) ↔ (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶)))
6 eleq1 2686 . . . . . . . . . 10 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
76anbi1d 740 . . . . . . . . 9 (𝑥 = 𝐶 → ((𝑥𝐴𝜑) ↔ (𝐶𝐴𝜑)))
87pm5.32ri 669 . . . . . . . 8 (((𝑥𝐴𝜑) ∧ 𝑥 = 𝐶) ↔ ((𝐶𝐴𝜑) ∧ 𝑥 = 𝐶))
95, 8bitr3i 266 . . . . . . 7 ((𝑥𝐴 ∧ (𝜑𝑥 = 𝐶)) ↔ ((𝐶𝐴𝜑) ∧ 𝑥 = 𝐶))
109anbi2i 729 . . . . . 6 ((𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶))) ↔ (𝑦𝐵 ∧ ((𝐶𝐴𝜑) ∧ 𝑥 = 𝐶)))
112, 4, 103bitr4ri 293 . . . . 5 ((𝑦𝐵 ∧ (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶))) ↔ (𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} ∧ 𝑥 = 𝐶))
1211rexbii2 3032 . . . 4 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶)) ↔ ∃𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝐶)
13 r19.42v 3084 . . . 4 (∃𝑦𝐵 (𝑥𝐴 ∧ (𝜑𝑥 = 𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 (𝜑𝑥 = 𝐶)))
14 nfrab1 3111 . . . . 5 𝑦{𝑦𝐵 ∣ (𝐶𝐴𝜑)}
15 nfcv 2761 . . . . 5 𝑧{𝑦𝐵 ∣ (𝐶𝐴𝜑)}
16 nfv 1840 . . . . 5 𝑧 𝑥 = 𝐶
17 nfcsb1v 3530 . . . . . 6 𝑦𝑧 / 𝑦𝐶
1817nfeq2 2776 . . . . 5 𝑦 𝑥 = 𝑧 / 𝑦𝐶
19 csbeq1a 3523 . . . . . 6 (𝑦 = 𝑧𝐶 = 𝑧 / 𝑦𝐶)
2019eqeq2d 2631 . . . . 5 (𝑦 = 𝑧 → (𝑥 = 𝐶𝑥 = 𝑧 / 𝑦𝐶))
2114, 15, 16, 18, 20cbvrexf 3154 . . . 4 (∃𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝐶 ↔ ∃𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶)
2212, 13, 213bitr3i 290 . . 3 ((𝑥𝐴 ∧ ∃𝑦𝐵 (𝜑𝑥 = 𝐶)) ↔ ∃𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶)
2322eubii 2491 . 2 (∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 (𝜑𝑥 = 𝐶)) ↔ ∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶)
24 elex 3198 . . . . . . . 8 (𝐶𝐴𝐶 ∈ V)
2524ad2antrl 763 . . . . . . 7 ((𝑦𝐵 ∧ (𝐶𝐴𝜑)) → 𝐶 ∈ V)
263, 25sylbi 207 . . . . . 6 (𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝐶 ∈ V)
2726rgen 2917 . . . . 5 𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝐶 ∈ V
28 nfv 1840 . . . . . 6 𝑧 𝐶 ∈ V
2917nfel1 2775 . . . . . 6 𝑦𝑧 / 𝑦𝐶 ∈ V
3019eleq1d 2683 . . . . . 6 (𝑦 = 𝑧 → (𝐶 ∈ V ↔ 𝑧 / 𝑦𝐶 ∈ V))
3114, 15, 28, 29, 30cbvralf 3153 . . . . 5 (∀𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝐶 ∈ V ↔ ∀𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑧 / 𝑦𝐶 ∈ V)
3227, 31mpbi 220 . . . 4 𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑧 / 𝑦𝐶 ∈ V
33 reusv2lem3 4831 . . . 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 2912 . . . . 5 (∀𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∀𝑧(𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝑧 / 𝑦𝐶))
36 nfv 1840 . . . . . 6 𝑧(𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶)
3714nfcri 2755 . . . . . . 7 𝑦 𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}
3837, 18nfim 1822 . . . . . 6 𝑦(𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝑧 / 𝑦𝐶)
39 eleq1 2686 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} ↔ 𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}))
4039, 20imbi12d 334 . . . . . 6 (𝑦 = 𝑧 → ((𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ (𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝑧 / 𝑦𝐶)))
4136, 38, 40cbval 2270 . . . . 5 (∀𝑦(𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑧(𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝑧 / 𝑦𝐶))
423imbi1i 339 . . . . . . . 8 ((𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ ((𝑦𝐵 ∧ (𝐶𝐴𝜑)) → 𝑥 = 𝐶))
43 impexp 462 . . . . . . . 8 (((𝑦𝐵 ∧ (𝐶𝐴𝜑)) → 𝑥 = 𝐶) ↔ (𝑦𝐵 → ((𝐶𝐴𝜑) → 𝑥 = 𝐶)))
4442, 43bitri 264 . . . . . . 7 ((𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ (𝑦𝐵 → ((𝐶𝐴𝜑) → 𝑥 = 𝐶)))
4544albii 1744 . . . . . 6 (∀𝑦(𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑦(𝑦𝐵 → ((𝐶𝐴𝜑) → 𝑥 = 𝐶)))
46 df-ral 2912 . . . . . 6 (∀𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶) ↔ ∀𝑦(𝑦𝐵 → ((𝐶𝐴𝜑) → 𝑥 = 𝐶)))
4745, 46bitr4i 267 . . . . 5 (∀𝑦(𝑦 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)} → 𝑥 = 𝐶) ↔ ∀𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
4835, 41, 473bitr2i 288 . . . 4 (∀𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∀𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
4948eubii 2491 . . 3 (∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
5034, 49bitri 264 . 2 (∃!𝑥𝑧 ∈ {𝑦𝐵 ∣ (𝐶𝐴𝜑)}𝑥 = 𝑧 / 𝑦𝐶 ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
511, 23, 503bitri 286 1 (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  ∃!weu 2469  wral 2907  wrex 2908  ∃!wreu 2909  {crab 2911  Vcvv 3186  csb 3514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749  ax-pow 4803
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-nul 3892
This theorem is referenced by:  reusv2lem5  4833
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