Theorem ra5 3131

Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1798. (Contributed by NM, 16-Jan-2004.)

Hypothesis

Ref	Expression
ra5.1	⊢ Ⅎxφ

Assertion

Ref	Expression
ra5	⊢ (∀x ∈ A (φ → ψ) → (φ → ∀x ∈ A ψ))

Proof of Theorem ra5

Step	Hyp	Ref	Expression
1		df-ral 2620	. . . 4 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
2		bi2.04 350	. . . . 5 ⊢ ((x ∈ A → (φ → ψ)) ↔ (φ → (x ∈ A → ψ)))
3	2	albii 1566	. . . 4 ⊢ (∀x(x ∈ A → (φ → ψ)) ↔ ∀x(φ → (x ∈ A → ψ)))
4	1, 3	bitri 240	. . 3 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(φ → (x ∈ A → ψ)))
5		ra5.1	. . . 4 ⊢ Ⅎxφ
6	5	stdpc5 1798	. . 3 ⊢ (∀x(φ → (x ∈ A → ψ)) → (φ → ∀x(x ∈ A → ψ)))
7	4, 6	sylbi 187	. 2 ⊢ (∀x ∈ A (φ → ψ) → (φ → ∀x(x ∈ A → ψ)))
8		df-ral 2620	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
9	7, 8	syl6ibr 218	1 ⊢ (∀x ∈ A (φ → ψ) → (φ → ∀x ∈ A ψ))

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-6 1729  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: (None)