Theorem alral 2520

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 1453	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		df-ral 2458	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4	2, 3	sylibr 134	1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1351   ∈ wcel 2146  ∀wral 2453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-gen 1447

This theorem depends on definitions:  df-bi 117  df-ral 2458

This theorem is referenced by:  abnex  4441  find  4592  prodeq2w  11532  findset  14266