Theorem r19.21v 2702

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

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

Distinct variable group:   φ,x

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

Proof of Theorem r19.21v

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2	1	r19.21 2701	1 ⊢ (∀x ∈ A (φ → ψ) ↔ (φ → ∀x ∈ A ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀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:  r19.32v  2758  rmo4  3030  2reu5lem3  3044  rmo3  3134