Theorem ralrid 3085

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 3078	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylibr 236	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1559   ∈ wcel 2143  ∀wral 3077

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 209  df-ral 3078

This theorem is referenced by:  alral  3092  hbralrimi  3153  axprlem4  5384  ingru  10784  bnj1476  35144  bnj1533  35149  bnj1523  35368  r1omhfb  35412  r1omhfbregs  35437  bj-axnul  37562  bj-axreprepsep  37565  exrecfnlem  37878  nninfnub  38255  eqab2  38754