Theorem ralimdvva 3192

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1810). (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 467	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))
3	2	ralimdva 3153	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐵 𝜒))
4	3	ralimdva 3153	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3053

This theorem is referenced by:  ralimdvvOLD  3195  dedekindle  11404  isdomn4  20681  islmhm2  21001  dflidl2rng  21184  dmatscmcl  22446  cpmatacl  22659  cpmatinvcl  22660  mat2pmatf1  22672  pmatcollpw2lem  22720  tgpt0  24062  isngp4  24556  addcnlem  24809  c1lip3  25961  aalioulem2  26298  aalioulem5  26301  aalioulem6  26302  aaliou  26303  iscgrglt  28498  2pthfrgrrn  30268  2pthfrgrrn2  30269  equivbnd  37819  ghomco  37920  fcoresf1  47065  fullthinc  49303