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Theorem brimralrspcev 5091
Description: Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
Assertion
Ref Expression
brimralrspcev ((𝐵𝑋 ∧ ∀𝑦𝑌 ((𝜑𝐴𝑅𝐵) → 𝜓)) → ∃𝑥𝑋𝑦𝑌 ((𝜑𝐴𝑅𝑥) → 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌   𝜑,𝑥   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑦)   𝐴(𝑦)   𝑅(𝑦)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem brimralrspcev
StepHypRef Expression
1 breq2 5034 . . 3 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21anbi2d 631 . 2 (𝑥 = 𝐵 → ((𝜑𝐴𝑅𝑥) ↔ (𝜑𝐴𝑅𝐵)))
32rspceaimv 3576 1 ((𝐵𝑋 ∧ ∀𝑦𝑌 ((𝜑𝐴𝑅𝐵) → 𝜓)) → ∃𝑥𝑋𝑦𝑌 ((𝜑𝐴𝑅𝑥) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wral 3106  wrex 3107   class class class wbr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031
This theorem is referenced by:  dveflem  24582  mullimc  42253  limcdm0  42255  mullimcf  42260  constlimc  42261  idlimc  42263  limcleqr  42281  addlimc  42285  0ellimcdiv  42286  ioodvbdlimc1lem2  42569  ioodvbdlimc2lem  42571
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