Theorem ralrimdvva 2709

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)

Hypothesis

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

Assertion

Ref	Expression
ralrimdvva	⊢ (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 ralrimdvva

Step	Hyp	Ref	Expression
1		ralrimdvva.1	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (ψ → χ))
2	1	ex 423	. . 3 ⊢ (φ → ((x ∈ A ∧ y ∈ B) → (ψ → χ)))
3	2	com23 72	. 2 ⊢ (φ → (ψ → ((x ∈ A ∧ y ∈ B) → χ)))
4	3	ralrimdvv 2708	1 ⊢ (φ → (ψ → ∀x ∈ A ∀y ∈ B χ))

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

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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: (None)