Theorem rspceaimv 3576

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 345	. . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))
3	2	ralbidv 3162	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐶 (𝜑 → 𝜒) ↔ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)))
4	3	rspcev 3571	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112

This theorem is referenced by:  brimralrspcev  5091  rexanre  14698  rexico  14705  rlim2lt  14846  rlim3  14847  rlimconst  14893  rlimcn2  14939  reccn2  14945  cn1lem  14946  o1rlimmul  14967  caucvgrlem  15021  divrcnv  15199  chfacffsupp  21461  chfacfscmulfsupp  21464  chfacfpmmulfsupp  21468  tsmsgsum  22744  tsmsres  22749  tsmsxp  22760  metcnpi3  23153  nrginvrcnlem  23297  nghmcn  23351  metdscn  23461  elcncf1di  23500  volcn  24210  itg2cnlem2  24366  abelthlem8  25034  divlogrlim  25226  cxplim  25557  cxploglim  25563  ftalem1  25658  ftalem2  25659  dchrisum0  26104  nmcvcn  28478  blocni  28588  0cnop  29762  0cnfn  29763  idcnop  29764  lnconi  29816  qqhcn  31342  dnicn  33944  ftc1anc  35138  limsupre3uzlem  42377  fourierdlem87  42835