Theorem alral 3154

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 1810	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		df-ral 3143	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	sylibr 236	1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1531   ∈ wcel 2110  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-ral 3143

This theorem is referenced by:  abnex  7478  find  7606  brdom5  9950  brdom4  9951  hashgt23el  13784  prodeq2w  15265  rpnnen2lem12  15577  umgr2cycllem  32387  umgr2cycl  32388  elpotr  33026  fvineqsnf1  34690  fvineqsneq  34692  phpreu  34875  ordelordALTVD  41199  rexrsb  43297