Theorem rspceaimv 3641

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 341	. . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))
3	2	ralbidv 3184	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐶 (𝜑 → 𝜒) ↔ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)))
4	3	rspcev 3635	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077

This theorem is referenced by:  brimralrspcev  5227  rexanre  15395  rexico  15402  rlim2lt  15543  rlim3  15544  rlimconst  15590  rlimcn3  15636  reccn2  15643  cn1lem  15644  o1rlimmul  15665  caucvgrlem  15721  divrcnv  15900  chfacffsupp  22883  chfacfscmulfsupp  22886  chfacfpmmulfsupp  22890  tsmsgsum  24168  tsmsres  24173  tsmsxp  24184  metcnpi3  24580  nrginvrcnlem  24733  nghmcn  24787  metdscn  24897  elcncf1di  24940  volcn  25660  itg2cnlem2  25817  abelthlem8  26501  divlogrlim  26695  cxplim  27033  cxploglim  27039  ftalem1  27134  ftalem2  27135  dchrisum0  27582  nmcvcn  30727  blocni  30837  0cnop  32011  0cnfn  32012  idcnop  32013  lnconi  32065  qqhcn  33937  dnicn  36458  ftc1anc  37661  limsupre3uzlem  45656  fourierdlem87  46114