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Theorem brimralrspcev 5205
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 5148 . . 3 (𝑥 = 𝐵 → (𝐴𝑅𝑥𝐴𝑅𝐵))
21anbi2d 628 . 2 (𝑥 = 𝐵 → ((𝜑𝐴𝑅𝑥) ↔ (𝜑𝐴𝑅𝐵)))
32rspceaimv 3609 1 ((𝐵𝑋 ∧ ∀𝑦𝑌 ((𝜑𝐴𝑅𝐵) → 𝜓)) → ∃𝑥𝑋𝑦𝑌 ((𝜑𝐴𝑅𝑥) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3051  wrex 3060   class class class wbr 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145
This theorem is referenced by:  dveflem  25924  mullimc  45063  limcdm0  45065  mullimcf  45070  constlimc  45071  idlimc  45073  limcleqr  45091  addlimc  45095  0ellimcdiv  45096  ioodvbdlimc1lem2  45379  ioodvbdlimc2lem  45381
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