Theorem ralrimdvv 2709

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem ralrimdvv

Step	Hyp	Ref	Expression
1		ralrimdvv.1	. . . 4 ⊢ (φ → (ψ → ((x ∈ A ∧ y ∈ B) → χ)))
2	1	imp 418	. . 3 ⊢ ((φ ∧ ψ) → ((x ∈ A ∧ y ∈ B) → χ))
3	2	ralrimivv 2706	. 2 ⊢ ((φ ∧ ψ) → ∀x ∈ A ∀y ∈ B χ)
4	3	ex 423	1 ⊢ (φ → (ψ → ∀x ∈ A ∀y ∈ B χ))

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2615

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 2620

This theorem is referenced by:  ralrimdvva  2710