Theorem alral 2673

Description: Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)

Assertion

Ref	Expression
alral	⊢ (∀xφ → ∀x ∈ A φ)

Proof of Theorem alral

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (φ → (x ∈ A → φ))
2	1	alimi 1559	. 2 ⊢ (∀xφ → ∀x(x ∈ A → φ))
3		df-ral 2620	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
4	2, 3	sylibr 203	1 ⊢ (∀xφ → ∀x ∈ A φ)

Syntax hints:   → wi 4  ∀wal 1540   ∈ 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

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by: (None)