Theorem rspceaimv 3628

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 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))
3	2	ralbidv 3197	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐶 (𝜑 → 𝜒) ↔ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)))
4	3	rspcev 3623	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814  df-clel 2893  df-ral 3143  df-rex 3144

This theorem is referenced by:  brimralrspcev  5127  rexanre  14706  rexico  14713  rlim2lt  14854  rlim3  14855  rlimconst  14901  rlimcn2  14947  reccn2  14953  cn1lem  14954  o1rlimmul  14975  caucvgrlem  15029  divrcnv  15207  chfacffsupp  21464  chfacfscmulfsupp  21467  chfacfpmmulfsupp  21471  tsmsgsum  22747  tsmsres  22752  tsmsxp  22763  metcnpi3  23156  nrginvrcnlem  23300  nghmcn  23354  metdscn  23464  elcncf1di  23503  volcn  24207  itg2cnlem2  24363  abelthlem8  25027  divlogrlim  25218  cxplim  25549  cxploglim  25555  ftalem1  25650  ftalem2  25651  dchrisum0  26096  nmcvcn  28472  blocni  28582  0cnop  29756  0cnfn  29757  idcnop  29758  lnconi  29810  qqhcn  31232  dnicn  33831  ftc1anc  34990  limsupre3uzlem  42065  fourierdlem87  42527