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Theorem rspceaimv 2821
 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 230 . . 3 (𝑥 = 𝐴 → ((𝜑𝜒) ↔ (𝜓𝜒)))
32ralbidv 2454 . 2 (𝑥 = 𝐴 → (∀𝑦𝐶 (𝜑𝜒) ↔ ∀𝑦𝐶 (𝜓𝜒)))
43rspcev 2813 1 ((𝐴𝐵 ∧ ∀𝑦𝐶 (𝜓𝜒)) → ∃𝑥𝐵𝑦𝐶 (𝜑𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711 This theorem is referenced by:  brimralrspcev  4019  reccn2ap  11187  metcnpi3  12856  elcncf1di  12905  mulcncflem  12929  limccnp2lem  12984
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