Theorem alral 2911

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		ala1 1754	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		df-ral 2900	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	sylibr 222	1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1472   ∈ wcel 1976  ∀wral 2895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727

This theorem depends on definitions:  df-bi 195  df-ral 2900

This theorem is referenced by:  find  6960  brdom5  9209  brdom4  9210  prodeq2w  14429  rpnnen2lem12  14741  elpotr  30723  phpreu  32346  neik0pk1imk0  37148  ordelordALTVD  37908  rexrsb  39601