Theorem ralrimivv 2516

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem ralrimivv

Step	Hyp	Ref	Expression
1		ralrimivv.1	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜓))
2	1	expd 256	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜓)))
3	2	ralrimdv 2514	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜓))
4	3	ralrimiv 2507	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1481  ∀wral 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2422

This theorem is referenced by:  ralrimivva  2517  ralrimdvv  2519  reuind  2893  ssrel2  4637  f1o2ndf1  6133  smoiso  6207  nndifsnid  6411  receuap  8454  lbreu  8727  tgcl  12272  topbas  12275  epttop  12298  restbasg  12376  txbas  12466  txbasval  12475  blfps  12617  blf  12618  blbas  12641