Theorem alral 3097

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 1777	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		df-ral 3086	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	sylibr 226	1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1506   ∈ wcel 2051  ∀wral 3081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773

This theorem depends on definitions:  df-bi 199  df-ral 3086

This theorem is referenced by:  abnex  7294  find  7420  brdom5  9747  brdom4  9748  hashgt23el  13596  prodeq2w  15124  rpnnen2lem12  15436  elpotr  32583  fvineqsnf1  34169  fvineqsneq  34171  phpreu  34354  neik0pk1imk0  39798  ordelordALTVD  40658  rexrsb  42738