Theorem ralrimd 2703

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)

Hypotheses

Ref	Expression
ralrimd.1	⊢ Ⅎxφ
ralrimd.2	⊢ Ⅎxψ
ralrimd.3	⊢ (φ → (ψ → (x ∈ A → χ)))

Assertion

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

Proof of Theorem ralrimd

Step	Hyp	Ref	Expression
1		ralrimd.1	. . 3 ⊢ Ⅎxφ
2		ralrimd.2	. . 3 ⊢ Ⅎxψ
3		ralrimd.3	. . 3 ⊢ (φ → (ψ → (x ∈ A → χ)))
4	1, 2, 3	alrimd 1769	. 2 ⊢ (φ → (ψ → ∀x(x ∈ A → χ)))
5		df-ral 2620	. 2 ⊢ (∀x ∈ A χ ↔ ∀x(x ∈ A → χ))
6	4, 5	syl6ibr 218	1 ⊢ (φ → (ψ → ∀x ∈ A χ))

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544   ∈ 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:  ralrimdv  2704  ncfinraise  4482