Theorem rspceaimv 3582

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 3159	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐶 (𝜑 → 𝜒) ↔ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)))
4	3	rspcev 3576	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061

This theorem is referenced by:  brimralrspcev  5159  rexanre  15270  rexico  15277  rlim2lt  15420  rlim3  15421  rlimconst  15467  rlimcn3  15513  reccn2  15520  cn1lem  15521  o1rlimmul  15542  caucvgrlem  15596  divrcnv  15775  chfacffsupp  22800  chfacfscmulfsupp  22803  chfacfpmmulfsupp  22807  tsmsgsum  24083  tsmsres  24088  tsmsxp  24099  metcnpi3  24490  nrginvrcnlem  24635  nghmcn  24689  metdscn  24801  elcncf1di  24844  volcn  25563  itg2cnlem2  25719  abelthlem8  26405  divlogrlim  26600  cxplim  26938  cxploglim  26944  ftalem1  27039  ftalem2  27040  dchrisum0  27487  nmcvcn  30770  blocni  30880  0cnop  32054  0cnfn  32055  idcnop  32056  lnconi  32108  qqhcn  34148  dnicn  36692  ftc1anc  37902  limsupre3uzlem  45979  fourierdlem87  46437