Theorem ralrid 3058

Description: Sufficient condition for the restricted universal quantifier. Deduction form. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypothesis

Ref	Expression
ralrid.1	⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Assertion

Ref	Expression
ralrid	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Proof of Theorem ralrid

Step	Hyp	Ref	Expression
1		ralrid.1	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
2		df-ral 3051	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylibr 235	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1541   ∈ wcel 2115  ∀wral 3050

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

This theorem depends on definitions:  df-bi 208  df-ral 3051

This theorem is referenced by:  alral  3065  hbralrimi  3126  axprlem4  5358  ingru  10732  bnj1476  35032  bnj1533  35037  bnj1523  35256  r1omhfb  35299  r1omhfbregs  35324  bj-axnul  37422  bj-axreprepsep  37425  exrecfnlem  37738  nninfnub  38115  eqab2  38614