Theorem rspceaimv 2919

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 231	. . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒)))
3	2	ralbidv 2533	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐶 (𝜑 → 𝜒) ↔ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)))
4	3	rspcev 2911	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝜑 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805

This theorem is referenced by:  brimralrspcev  4153  reccn2ap  11936  mplsubgfilemm  14782  mplsubgfilemcl  14783  metcnpi3  15311  elcncf1di  15373  mulcncflem  15401  limccnp2lem  15470