Theorem brimralrspcev 5165

Description: Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)

Assertion

Ref	Expression
brimralrspcev	⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜓)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝑥) → 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌   𝜑,𝑥   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑦)   𝐴(𝑦)   𝑅(𝑦)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem brimralrspcev

Step	Hyp	Ref	Expression
1		breq2 5106	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
2	1	anbi2d 642	. 2 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝐴𝑅𝑥) ↔ (𝜑 ∧ 𝐴𝑅𝐵)))
3	2	rspceaimv 3582	1 ⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜓)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝑥) → 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   class class class wbr 5102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103

This theorem is used by:  dveflem  26260  mullimc  46550  limcdm0  46552  mullimcf  46557  constlimc  46558  idlimc  46560  limcleqr  46576  addlimc  46580  0ellimcdiv  46581  ioodvbdlimc1lem2  46864  ioodvbdlimc2lem  46866