Theorem alral 3156

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 3145	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	sylibr 236	1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1535   ∈ wcel 2114  ∀wral 3140

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 209  df-ral 3145

This theorem is referenced by:  abnex  7481  find  7609  brdom5  9953  brdom4  9954  hashgt23el  13788  prodeq2w  15268  rpnnen2lem12  15580  umgr2cycllem  32389  umgr2cycl  32390  elpotr  33028  fvineqsnf1  34693  fvineqsneq  34695  phpreu  34878  ordelordALTVD  41208  rexrsb  43305