Theorem brimralrspcev 5126

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 5069	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
2	1	anbi2d 630	. 2 ⊢ (𝑥 = 𝐵 → ((𝜑 ∧ 𝐴𝑅𝑥) ↔ (𝜑 ∧ 𝐴𝑅𝐵)))
3	2	rspceaimv 3627	1 ⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜓)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝑥) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   class class class wbr 5065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066

This theorem is referenced by:  dveflem  24575  mullimc  41895  limcdm0  41897  mullimcf  41902  constlimc  41903  idlimc  41905  limcleqr  41923  addlimc  41927  0ellimcdiv  41928  ioodvbdlimc1lem2  42215  ioodvbdlimc2lem  42217