Theorem rspceaimv 3566

Description: Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)

Hypothesis

Ref	Expression
rspceaimv.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rspceaimv	⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))

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

Step	Hyp	Ref	Expression
1		rspceaimv.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	imbi1d 342	. . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))
3	2	ralbidv 3113	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐶 (𝜑 → 𝜒) ↔ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)))
4	3	rspcev 3562	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2107  ∀wral 3065  ∃wrex 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2069  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071

This theorem is referenced by:  brimralrspcev  5136  rexanre  15067  rexico  15074  rlim2lt  15215  rlim3  15216  rlimconst  15262  rlimcn3  15308  reccn2  15315  cn1lem  15316  o1rlimmul  15337  caucvgrlem  15393  divrcnv  15573  chfacffsupp  22014  chfacfscmulfsupp  22017  chfacfpmmulfsupp  22021  tsmsgsum  23299  tsmsres  23304  tsmsxp  23315  metcnpi3  23711  nrginvrcnlem  23864  nghmcn  23918  metdscn  24028  elcncf1di  24067  volcn  24779  itg2cnlem2  24936  abelthlem8  25607  divlogrlim  25799  cxplim  26130  cxploglim  26136  ftalem1  26231  ftalem2  26232  dchrisum0  26677  nmcvcn  29066  blocni  29176  0cnop  30350  0cnfn  30351  idcnop  30352  lnconi  30404  qqhcn  31950  dnicn  34681  ftc1anc  35867  limsupre3uzlem  43283  fourierdlem87  43741