Theorem alral 3081

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 1811	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		df-ral 3068	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3	1, 2	sylibr 234	1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1535   ∈ wcel 2108  ∀wral 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807

This theorem depends on definitions:  df-bi 207  df-ral 3068

This theorem is referenced by:  abnex  7792  find  7935  brdom5  10598  brdom4  10599  hashgt23el  14473  prodeq2w  15958  rpnnen2lem12  16273  umgr2cycllem  35108  umgr2cycl  35109  elpotr  35745  fvineqsnf1  37376  fvineqsneq  37378  phpreu  37564  ordelordALTVD  44838  rexrsb  47015