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Theorem brimralrspcev 5203
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 5146 . . 3 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21anbi2d 628 . 2 (𝑥 = 𝐵 → ((𝜑𝐴𝑅𝑥) ↔ (𝜑𝐴𝑅𝐵)))
32rspceaimv 3613 1 ((𝐵𝑋 ∧ ∀𝑦𝑌 ((𝜑𝐴𝑅𝐵) → 𝜓)) → ∃𝑥𝑋𝑦𝑌 ((𝜑𝐴𝑅𝑥) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3056  wrex 3065   class class class wbr 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143
This theorem is referenced by:  dveflem  25898  mullimc  44927  limcdm0  44929  mullimcf  44934  constlimc  44935  idlimc  44937  limcleqr  44955  addlimc  44959  0ellimcdiv  44960  ioodvbdlimc1lem2  45243  ioodvbdlimc2lem  45245
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