Theorem ralrimdva 2705

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

Hypothesis

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

Assertion

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

Distinct variable groups:   φ,x   ψ,x

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

Proof of Theorem ralrimdva

Step	Hyp	Ref	Expression
1		ralrimdva.1	. . . 4 ⊢ ((φ ∧ x ∈ A) → (ψ → χ))
2	1	ex 423	. . 3 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	2	com23 72	. 2 ⊢ (φ → (ψ → (x ∈ A → χ)))
4	3	ralrimdv 2704	1 ⊢ (φ → (ψ → ∀x ∈ A χ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2615

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is used by: (None)