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Theorem bnd2 3926
 Description: A variant of the Boundedness Axiom bnd 3925 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.)
Hypothesis
Ref Expression
bnd2.1 𝐴 ∈ V
Assertion
Ref Expression
bnd2 (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))
Distinct variable groups:   𝜑,𝑧   𝑥,𝑧,𝐴   𝑥,𝑦,𝐵,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem bnd2
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2312 . . . 4 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
21ralbii 2330 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝜑))
3 bnd2.1 . . . 4 𝐴 ∈ V
4 raleq 2505 . . . . 5 (𝑣 = 𝐴 → (∀𝑥𝑣𝑦(𝑦𝐵𝜑) ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝜑)))
5 raleq 2505 . . . . . 6 (𝑣 = 𝐴 → (∀𝑥𝑣𝑦𝑤 (𝑦𝐵𝜑) ↔ ∀𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑)))
65exbidv 1706 . . . . 5 (𝑣 = 𝐴 → (∃𝑤𝑥𝑣𝑦𝑤 (𝑦𝐵𝜑) ↔ ∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑)))
74, 6imbi12d 223 . . . 4 (𝑣 = 𝐴 → ((∀𝑥𝑣𝑦(𝑦𝐵𝜑) → ∃𝑤𝑥𝑣𝑦𝑤 (𝑦𝐵𝜑)) ↔ (∀𝑥𝐴𝑦(𝑦𝐵𝜑) → ∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑))))
8 bnd 3925 . . . 4 (∀𝑥𝑣𝑦(𝑦𝐵𝜑) → ∃𝑤𝑥𝑣𝑦𝑤 (𝑦𝐵𝜑))
93, 7, 8vtocl 2608 . . 3 (∀𝑥𝐴𝑦(𝑦𝐵𝜑) → ∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑))
102, 9sylbi 114 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑))
11 vex 2560 . . . . 5 𝑤 ∈ V
1211inex1 3891 . . . 4 (𝑤𝐵) ∈ V
13 inss2 3158 . . . . . . 7 (𝑤𝐵) ⊆ 𝐵
14 sseq1 2966 . . . . . . 7 (𝑧 = (𝑤𝐵) → (𝑧𝐵 ↔ (𝑤𝐵) ⊆ 𝐵))
1513, 14mpbiri 157 . . . . . 6 (𝑧 = (𝑤𝐵) → 𝑧𝐵)
1615biantrurd 289 . . . . 5 (𝑧 = (𝑤𝐵) → (∀𝑥𝐴𝑦𝑧 𝜑 ↔ (𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑)))
17 rexeq 2506 . . . . . . 7 (𝑧 = (𝑤𝐵) → (∃𝑦𝑧 𝜑 ↔ ∃𝑦 ∈ (𝑤𝐵)𝜑))
18 elin 3126 . . . . . . . . . 10 (𝑦 ∈ (𝑤𝐵) ↔ (𝑦𝑤𝑦𝐵))
1918anbi1i 431 . . . . . . . . 9 ((𝑦 ∈ (𝑤𝐵) ∧ 𝜑) ↔ ((𝑦𝑤𝑦𝐵) ∧ 𝜑))
20 anass 381 . . . . . . . . 9 (((𝑦𝑤𝑦𝐵) ∧ 𝜑) ↔ (𝑦𝑤 ∧ (𝑦𝐵𝜑)))
2119, 20bitri 173 . . . . . . . 8 ((𝑦 ∈ (𝑤𝐵) ∧ 𝜑) ↔ (𝑦𝑤 ∧ (𝑦𝐵𝜑)))
2221rexbii2 2335 . . . . . . 7 (∃𝑦 ∈ (𝑤𝐵)𝜑 ↔ ∃𝑦𝑤 (𝑦𝐵𝜑))
2317, 22syl6bb 185 . . . . . 6 (𝑧 = (𝑤𝐵) → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑤 (𝑦𝐵𝜑)))
2423ralbidv 2326 . . . . 5 (𝑧 = (𝑤𝐵) → (∀𝑥𝐴𝑦𝑧 𝜑 ↔ ∀𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑)))
2516, 24bitr3d 179 . . . 4 (𝑧 = (𝑤𝐵) → ((𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑) ↔ ∀𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑)))
2612, 25spcev 2647 . . 3 (∀𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑) → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))
2726exlimiv 1489 . 2 (∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑) → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))
2810, 27syl 14 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  Vcvv 2557   ∩ cin 2916   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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