Theorem alral 1689

Description: Universal quantification implies restricted quantification.

Assertion

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

Proof of Theorem alral

Step	Hyp	Ref	Expression
1		ax-1 4	. . 3 ⊢ (φ → (x ∈ A → φ))
2	1	19.20i 990	. 2 ⊢ (∀xφ → ∀x(x ∈ A → φ))
3		df-ral 1646	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
4	2, 3	sylibr 200	1 ⊢ (∀xφ → ∀x ∈ A φ)

Syntax hints:   → wi 3  ∀wal 952   ∈ wcel 956  ∀wral 1642

This theorem is referenced by:  brdom5 4782  brdom4 4783  gelcomplOLD 10353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-ral 1646

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