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Theorem bnd2 3953
Description: A variant of the Boundedness Axiom bnd 3952 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 2329 . . . 4 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
21ralbii 2347 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝜑))
3 bnd2.1 . . . 4 𝐴 ∈ V
4 raleq 2522 . . . . 5 (𝑣 = 𝐴 → (∀𝑥𝑣𝑦(𝑦𝐵𝜑) ↔ ∀𝑥𝐴𝑦(𝑦𝐵𝜑)))
5 raleq 2522 . . . . . 6 (𝑣 = 𝐴 → (∀𝑥𝑣𝑦𝑤 (𝑦𝐵𝜑) ↔ ∀𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑)))
65exbidv 1722 . . . . 5 (𝑣 = 𝐴 → (∃𝑤𝑥𝑣𝑦𝑤 (𝑦𝐵𝜑) ↔ ∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑)))
74, 6imbi12d 227 . . . 4 (𝑣 = 𝐴 → ((∀𝑥𝑣𝑦(𝑦𝐵𝜑) → ∃𝑤𝑥𝑣𝑦𝑤 (𝑦𝐵𝜑)) ↔ (∀𝑥𝐴𝑦(𝑦𝐵𝜑) → ∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑))))
8 bnd 3952 . . . 4 (∀𝑥𝑣𝑦(𝑦𝐵𝜑) → ∃𝑤𝑥𝑣𝑦𝑤 (𝑦𝐵𝜑))
93, 7, 8vtocl 2625 . . 3 (∀𝑥𝐴𝑦(𝑦𝐵𝜑) → ∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑))
102, 9sylbi 118 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑤𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑))
11 vex 2577 . . . . 5 𝑤 ∈ V
1211inex1 3918 . . . 4 (𝑤𝐵) ∈ V
13 inss2 3185 . . . . . . 7 (𝑤𝐵) ⊆ 𝐵
14 sseq1 2993 . . . . . . 7 (𝑧 = (𝑤𝐵) → (𝑧𝐵 ↔ (𝑤𝐵) ⊆ 𝐵))
1513, 14mpbiri 161 . . . . . 6 (𝑧 = (𝑤𝐵) → 𝑧𝐵)
1615biantrurd 293 . . . . 5 (𝑧 = (𝑤𝐵) → (∀𝑥𝐴𝑦𝑧 𝜑 ↔ (𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑)))
17 rexeq 2523 . . . . . . 7 (𝑧 = (𝑤𝐵) → (∃𝑦𝑧 𝜑 ↔ ∃𝑦 ∈ (𝑤𝐵)𝜑))
18 elin 3153 . . . . . . . . . 10 (𝑦 ∈ (𝑤𝐵) ↔ (𝑦𝑤𝑦𝐵))
1918anbi1i 439 . . . . . . . . 9 ((𝑦 ∈ (𝑤𝐵) ∧ 𝜑) ↔ ((𝑦𝑤𝑦𝐵) ∧ 𝜑))
20 anass 387 . . . . . . . . 9 (((𝑦𝑤𝑦𝐵) ∧ 𝜑) ↔ (𝑦𝑤 ∧ (𝑦𝐵𝜑)))
2119, 20bitri 177 . . . . . . . 8 ((𝑦 ∈ (𝑤𝐵) ∧ 𝜑) ↔ (𝑦𝑤 ∧ (𝑦𝐵𝜑)))
2221rexbii2 2352 . . . . . . 7 (∃𝑦 ∈ (𝑤𝐵)𝜑 ↔ ∃𝑦𝑤 (𝑦𝐵𝜑))
2317, 22syl6bb 189 . . . . . 6 (𝑧 = (𝑤𝐵) → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑤 (𝑦𝐵𝜑)))
2423ralbidv 2343 . . . . 5 (𝑧 = (𝑤𝐵) → (∀𝑥𝐴𝑦𝑧 𝜑 ↔ ∀𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑)))
2516, 24bitr3d 183 . . . 4 (𝑧 = (𝑤𝐵) → ((𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑) ↔ ∀𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑)))
2612, 25spcev 2664 . . 3 (∀𝑥𝐴𝑦𝑤 (𝑦𝐵𝜑) → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))
2726exlimiv 1505 . 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 101   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  Vcvv 2574  cin 2943  wss 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2951  df-ss 2958
This theorem is referenced by: (None)
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