Theorem alral 3065

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 1814	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		df-ral 3052	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	sylibr 234	1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2113  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-ral 3052

This theorem is referenced by:  falseral0  4467  abnex  7702  find  7837  brdom5  10439  brdom4  10440  hashgt23el  14347  prodeq2w  15833  rpnnen2lem12  16150  umgr2cycllem  35334  umgr2cycl  35335  elpotr  35973  fvineqsnf1  37611  fvineqsneq  37613  phpreu  37801  ordelordALTVD  45103  ssclaxsep  45219  rexrsb  47342