Theorem ralimdvva 3209

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1843). (Contributed by AV, 27-Nov-2019.)

Hypothesis

Ref	Expression
ralimdvva.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))

Assertion

Ref	Expression
ralimdvva	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝜑

Allowed substitution hints:   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)   𝐴(𝑥)   𝐵(𝑥, 𝑦)

Proof of Theorem ralimdvva

Step	Hyp	Ref	Expression
1		ralimdvva.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))
2	1	anassrs 473	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	ralimdva 3174	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐵 𝜒))
4	3	ralimdva 3174	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3076

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

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3077

This theorem is used by:  ralimdvvOLD  3212  dedekindle  11401  isdomn4  20880  islmhm2  21225  dflidl2rng  21409  dmatscmcl  22728  cpmatacl  22944  cpmatinvcl  22945  mat2pmatf1  22957  pmatcollpw2lem  23005  tgpt0  24348  isngp4  24841  addcnlem  25094  c1lip3  26229  aalioulem2  26572  aalioulem5  26575  aalioulem6  26576  aaliou  26577  iscgrglt  28859  2pthfrgrrn  30765  2pthfrgrrn2  30766  equivbnd  38543  ghomco  38644  fcoresf1  47960  fullthinc  50379