Theorem ralrimivv 2705

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	exp3a 425	. . 3 ⊢ (φ → (x ∈ A → (y ∈ B → ψ)))
3	2	ralrimdv 2703	. 2 ⊢ (φ → (x ∈ A → ∀y ∈ B ψ))
4	3	ralrimiv 2696	1 ⊢ (φ → ∀x ∈ A ∀y ∈ B ψ)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619

This theorem is referenced by:  ralrimivva  2706  ralrimdvv  2708  reuind  3039  nnpweq  4523  trrd  5925  connexrd  5930