Theorem ralrimivv 2631

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 258	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜓)))
3	2	ralrimdv 2629	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜓))
4	3	ralrimiv 2622	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∈ wcel 2209  ∀wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is used by:  ralrimivva  2632  ralrimdvv  2634  reuind  3031  ssrel2  4867  f1o2ndf1  6468  smoiso  6577  nndifsnid  6784  receuap  9007  lbreu  9283  0subm  13850  insubm  13851  iscmnd  14191  quscrng  14979  tgcl  15290  topbas  15293  epttop  15316  restbasg  15394  txbas  15484  txbasval  15493  blfps  15635  blf  15636  blbas  15659