ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brralrspcev GIF version

Theorem brralrspcev 4047
Description: Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
Assertion
Ref Expression
brralrspcev ((𝐵𝑋 ∧ ∀𝑦𝑌 𝐴𝑅𝐵) → ∃𝑥𝑋𝑦𝑌 𝐴𝑅𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑦)   𝑅(𝑦)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem brralrspcev
StepHypRef Expression
1 breq2 3993 . . 3 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21ralbidv 2470 . 2 (𝑥 = 𝐵 → (∀𝑦𝑌 𝐴𝑅𝑥 ↔ ∀𝑦𝑌 𝐴𝑅𝐵))
32rspcev 2834 1 ((𝐵𝑋 ∧ ∀𝑦𝑌 𝐴𝑅𝐵) → ∃𝑥𝑋𝑦𝑌 𝐴𝑅𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  wral 2448  wrex 2449   class class class wbr 3989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990
This theorem is referenced by:  axpre-suploclemres  7863  fiubm  10763  dedekindeulemub  13390  suplociccreex  13396  dedekindicclemub  13399
  Copyright terms: Public domain W3C validator