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Theorem rspceaimv 2849
Description: Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)
Hypothesis
Ref Expression
rspceaimv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspceaimv ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem rspceaimv
StepHypRef Expression
1 rspceaimv.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
21imbi1d 231 . . 3 (𝑥 = 𝐴 → ((𝜑𝜒) ↔ (𝜓𝜒)))
32ralbidv 2477 . 2 (𝑥 = 𝐴 → (∀𝑦𝐶 (𝜑𝜒) ↔ ∀𝑦𝐶 (𝜓𝜒)))
43rspcev 2841 1 ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739
This theorem is referenced by:  brimralrspcev  4061  reccn2ap  11314  metcnpi3  13888  elcncf1di  13937  mulcncflem  13961  limccnp2lem  14016
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