Theorem r19.21 2701

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)

Hypothesis

Ref	Expression
r19.21.1	⊢ Ⅎxφ

Assertion

Ref	Expression
r19.21	⊢ (∀x ∈ A (φ → ψ) ↔ (φ → ∀x ∈ A ψ))

Proof of Theorem r19.21

Step	Hyp	Ref	Expression
1		r19.21.1	. 2 ⊢ Ⅎxφ
2		r19.21t 2700	. 2 ⊢ (Ⅎxφ → (∀x ∈ A (φ → ψ) ↔ (φ → ∀x ∈ A ψ)))
3	1, 2	ax-mp 5	1 ⊢ (∀x ∈ A (φ → ψ) ↔ (φ → ∀x ∈ A ψ))

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  ∀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:  r19.21v  2702