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Theorem unirep 33812
Description: Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.)
Hypotheses
Ref Expression
unirep.1 (𝑦 = 𝐷 → (𝜑𝜓))
unirep.2 (𝑦 = 𝐷𝐵 = 𝐶)
unirep.3 (𝑦 = 𝑧 → (𝜑𝜒))
unirep.4 (𝑦 = 𝑧𝐵 = 𝐹)
unirep.5 𝐵 ∈ V
Assertion
Ref Expression
unirep ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → (℩𝑥𝑦𝐴 (𝜑𝑥 = 𝐵)) = 𝐶)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑧   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦   𝜑,𝑥,𝑧   𝜓,𝑥,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑧)   𝜒(𝑧)   𝐵(𝑦)   𝐶(𝑧)   𝐷(𝑧)   𝐹(𝑧)

Proof of Theorem unirep
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2753 . . . . 5 (𝜓𝐶 = 𝐶)
21ancli 575 . . . 4 (𝜓 → (𝜓𝐶 = 𝐶))
3 unirep.1 . . . . . 6 (𝑦 = 𝐷 → (𝜑𝜓))
4 unirep.2 . . . . . . 7 (𝑦 = 𝐷𝐵 = 𝐶)
54eqeq2d 2762 . . . . . 6 (𝑦 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
63, 5anbi12d 749 . . . . 5 (𝑦 = 𝐷 → ((𝜑𝐶 = 𝐵) ↔ (𝜓𝐶 = 𝐶)))
76rspcev 3441 . . . 4 ((𝐷𝐴 ∧ (𝜓𝐶 = 𝐶)) → ∃𝑦𝐴 (𝜑𝐶 = 𝐵))
82, 7sylan2 492 . . 3 ((𝐷𝐴𝜓) → ∃𝑦𝐴 (𝜑𝐶 = 𝐵))
98adantl 473 . 2 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → ∃𝑦𝐴 (𝜑𝐶 = 𝐵))
10 nfcvd 2895 . . . . . 6 (𝐷𝐴𝑦𝐶)
1110, 4csbiegf 3690 . . . . 5 (𝐷𝐴𝐷 / 𝑦𝐵 = 𝐶)
12 unirep.5 . . . . . 6 𝐵 ∈ V
1312csbex 4937 . . . . 5 𝐷 / 𝑦𝐵 ∈ V
1411, 13syl6eqelr 2840 . . . 4 (𝐷𝐴𝐶 ∈ V)
1514ad2antrl 766 . . 3 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → 𝐶 ∈ V)
16 eqeq1 2756 . . . . . . . . . . 11 (𝑥 = 𝐶 → (𝑥 = 𝐵𝐶 = 𝐵))
1716anbi2d 742 . . . . . . . . . 10 (𝑥 = 𝐶 → ((𝜑𝑥 = 𝐵) ↔ (𝜑𝐶 = 𝐵)))
1817rexbidv 3182 . . . . . . . . 9 (𝑥 = 𝐶 → (∃𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑𝐶 = 𝐵)))
1918spcegv 3426 . . . . . . . 8 (𝐶 ∈ V → (∃𝑦𝐴 (𝜑𝐶 = 𝐵) → ∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵)))
2014, 19syl 17 . . . . . . 7 (𝐷𝐴 → (∃𝑦𝐴 (𝜑𝐶 = 𝐵) → ∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵)))
2120adantr 472 . . . . . 6 ((𝐷𝐴𝜓) → (∃𝑦𝐴 (𝜑𝐶 = 𝐵) → ∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵)))
228, 21mpd 15 . . . . 5 ((𝐷𝐴𝜓) → ∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵))
2322adantl 473 . . . 4 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → ∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵))
24 r19.29 3202 . . . . . . . 8 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ ∃𝑦𝐴 (𝜑𝑥 = 𝐵)) → ∃𝑦𝐴 (∀𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜑𝑥 = 𝐵)))
25 r19.29 3202 . . . . . . . . . . . 12 ((∀𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ ∃𝑧𝐴 (𝜒𝑤 = 𝐹)) → ∃𝑧𝐴 (((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜒𝑤 = 𝐹)))
26 an4 900 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 = 𝐵) ∧ (𝜒𝑤 = 𝐹)) ↔ ((𝜑𝜒) ∧ (𝑥 = 𝐵𝑤 = 𝐹)))
27 pm3.35 612 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝜒) ∧ ((𝜑𝜒) → 𝐵 = 𝐹)) → 𝐵 = 𝐹)
28 eqeq12 2765 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝐵𝑤 = 𝐹) → (𝑥 = 𝑤𝐵 = 𝐹))
2927, 28syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝜒) ∧ ((𝜑𝜒) → 𝐵 = 𝐹)) → ((𝑥 = 𝐵𝑤 = 𝐹) → 𝑥 = 𝑤))
3029ancoms 468 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜑𝜒)) → ((𝑥 = 𝐵𝑤 = 𝐹) → 𝑥 = 𝑤))
3130expimpd 630 . . . . . . . . . . . . . . . . 17 (((𝜑𝜒) → 𝐵 = 𝐹) → (((𝜑𝜒) ∧ (𝑥 = 𝐵𝑤 = 𝐹)) → 𝑥 = 𝑤))
3226, 31syl5bi 232 . . . . . . . . . . . . . . . 16 (((𝜑𝜒) → 𝐵 = 𝐹) → (((𝜑𝑥 = 𝐵) ∧ (𝜒𝑤 = 𝐹)) → 𝑥 = 𝑤))
3332ancomsd 469 . . . . . . . . . . . . . . 15 (((𝜑𝜒) → 𝐵 = 𝐹) → (((𝜒𝑤 = 𝐹) ∧ (𝜑𝑥 = 𝐵)) → 𝑥 = 𝑤))
3433expdimp 452 . . . . . . . . . . . . . 14 ((((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜒𝑤 = 𝐹)) → ((𝜑𝑥 = 𝐵) → 𝑥 = 𝑤))
3534rexlimivw 3159 . . . . . . . . . . . . 13 (∃𝑧𝐴 (((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜒𝑤 = 𝐹)) → ((𝜑𝑥 = 𝐵) → 𝑥 = 𝑤))
3635imp 444 . . . . . . . . . . . 12 ((∃𝑧𝐴 (((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜒𝑤 = 𝐹)) ∧ (𝜑𝑥 = 𝐵)) → 𝑥 = 𝑤)
3725, 36sylan 489 . . . . . . . . . . 11 (((∀𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ ∃𝑧𝐴 (𝜒𝑤 = 𝐹)) ∧ (𝜑𝑥 = 𝐵)) → 𝑥 = 𝑤)
3837an32s 881 . . . . . . . . . 10 (((∀𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜑𝑥 = 𝐵)) ∧ ∃𝑧𝐴 (𝜒𝑤 = 𝐹)) → 𝑥 = 𝑤)
3938ex 449 . . . . . . . . 9 ((∀𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜑𝑥 = 𝐵)) → (∃𝑧𝐴 (𝜒𝑤 = 𝐹) → 𝑥 = 𝑤))
4039rexlimivw 3159 . . . . . . . 8 (∃𝑦𝐴 (∀𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝜑𝑥 = 𝐵)) → (∃𝑧𝐴 (𝜒𝑤 = 𝐹) → 𝑥 = 𝑤))
4124, 40syl 17 . . . . . . 7 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ ∃𝑦𝐴 (𝜑𝑥 = 𝐵)) → (∃𝑧𝐴 (𝜒𝑤 = 𝐹) → 𝑥 = 𝑤))
4241expimpd 630 . . . . . 6 (∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) → ((∃𝑦𝐴 (𝜑𝑥 = 𝐵) ∧ ∃𝑧𝐴 (𝜒𝑤 = 𝐹)) → 𝑥 = 𝑤))
4342adantr 472 . . . . 5 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → ((∃𝑦𝐴 (𝜑𝑥 = 𝐵) ∧ ∃𝑧𝐴 (𝜒𝑤 = 𝐹)) → 𝑥 = 𝑤))
4443alrimivv 1997 . . . 4 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → ∀𝑥𝑤((∃𝑦𝐴 (𝜑𝑥 = 𝐵) ∧ ∃𝑧𝐴 (𝜒𝑤 = 𝐹)) → 𝑥 = 𝑤))
45 eqeq1 2756 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 = 𝐵𝑤 = 𝐵))
4645anbi2d 742 . . . . . . 7 (𝑥 = 𝑤 → ((𝜑𝑥 = 𝐵) ↔ (𝜑𝑤 = 𝐵)))
4746rexbidv 3182 . . . . . 6 (𝑥 = 𝑤 → (∃𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑𝑤 = 𝐵)))
48 unirep.3 . . . . . . . 8 (𝑦 = 𝑧 → (𝜑𝜒))
49 unirep.4 . . . . . . . . 9 (𝑦 = 𝑧𝐵 = 𝐹)
5049eqeq2d 2762 . . . . . . . 8 (𝑦 = 𝑧 → (𝑤 = 𝐵𝑤 = 𝐹))
5148, 50anbi12d 749 . . . . . . 7 (𝑦 = 𝑧 → ((𝜑𝑤 = 𝐵) ↔ (𝜒𝑤 = 𝐹)))
5251cbvrexv 3303 . . . . . 6 (∃𝑦𝐴 (𝜑𝑤 = 𝐵) ↔ ∃𝑧𝐴 (𝜒𝑤 = 𝐹))
5347, 52syl6bb 276 . . . . 5 (𝑥 = 𝑤 → (∃𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑧𝐴 (𝜒𝑤 = 𝐹)))
5453eu4 2648 . . . 4 (∃!𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ∧ ∀𝑥𝑤((∃𝑦𝐴 (𝜑𝑥 = 𝐵) ∧ ∃𝑧𝐴 (𝜒𝑤 = 𝐹)) → 𝑥 = 𝑤)))
5523, 44, 54sylanbrc 701 . . 3 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → ∃!𝑥𝑦𝐴 (𝜑𝑥 = 𝐵))
5618iota2 6030 . . 3 ((𝐶 ∈ V ∧ ∃!𝑥𝑦𝐴 (𝜑𝑥 = 𝐵)) → (∃𝑦𝐴 (𝜑𝐶 = 𝐵) ↔ (℩𝑥𝑦𝐴 (𝜑𝑥 = 𝐵)) = 𝐶))
5715, 55, 56syl2anc 696 . 2 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → (∃𝑦𝐴 (𝜑𝐶 = 𝐵) ↔ (℩𝑥𝑦𝐴 (𝜑𝑥 = 𝐵)) = 𝐶))
589, 57mpbid 222 1 ((∀𝑦𝐴𝑧𝐴 ((𝜑𝜒) → 𝐵 = 𝐹) ∧ (𝐷𝐴𝜓)) → (℩𝑥𝑦𝐴 (𝜑𝑥 = 𝐵)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1622   = wceq 1624  wex 1845  wcel 2131  ∃!weu 2599  wral 3042  wrex 3043  Vcvv 3332  csb 3666  cio 6002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-nul 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-nul 4051  df-sn 4314  df-pr 4316  df-uni 4581  df-iota 6004
This theorem is referenced by: (None)
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