Theorem rax5 1472

Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 59. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 739.

Hypothesis

Ref	Expression
rax5.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
rax5	⊢ (∀x ∈ A (φ → ψ) → (φ → ∀x ∈ A ψ))

Proof of Theorem rax5

Step	Hyp	Ref	Expression
1		df-ral 1205	. . . 4 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
2		bi2.04 141	. . . . 5 ⊢ ((x ∈ A → (φ → ψ)) ↔ (φ → (x ∈ A → ψ)))
3	2	bial 695	. . . 4 ⊢ (∀x(x ∈ A → (φ → ψ)) ↔ ∀x(φ → (x ∈ A → ψ)))
4	1, 3	bitr 151	. . 3 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(φ → (x ∈ A → ψ)))
5		rax5.1	. . . 4 ⊢ (φ → ∀xφ)
6	5	stdpc5 739	. . 3 ⊢ (∀x(φ → (x ∈ A → ψ)) → (φ → ∀x(x ∈ A → ψ)))
7	4, 6	sylbi 174	. 2 ⊢ (∀x ∈ A (φ → ψ) → (φ → ∀x(x ∈ A → ψ)))
8		df-ral 1205	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
9	7, 8	syl6ibr 186	1 ⊢ (∀x ∈ A (φ → ψ) → (φ → ∀x ∈ A ψ))

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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