Theorem ralrimdv 2704

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)

Hypothesis

Ref	Expression
ralrimdv.1	⊢ (φ → (ψ → (x ∈ A → χ)))

Assertion

Ref	Expression
ralrimdv	⊢ (φ → (ψ → ∀x ∈ A χ))

Distinct variable groups:   φ,x   ψ,x

Allowed substitution hints:   χ(x)   A(x)

Proof of Theorem ralrimdv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		nfv 1619	. 2 ⊢ Ⅎxψ
3		ralrimdv.1	. 2 ⊢ (φ → (ψ → (x ∈ A → χ)))
4	1, 2, 3	ralrimd 2703	1 ⊢ (φ → (ψ → ∀x ∈ A χ))

Syntax hints:   → wi 4   ∈ 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-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  ralrimdva  2705  ralrimivv  2706  nndisjeq  4430  trrd  5926