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 237	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1566   ∈ wcel 2141  ∀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 210  df-ral 3078

This theorem is referenced by:  alral  3092  hbralrimi  3153  axprlem4  5397  ingru  10799  bnj1476  35201  bnj1533  35206  bnj1523  35425  r1omhfb  35470  r1omhfbregs  35504  bj-axnul  37653  bj-axreprepsep  37656  exrecfnlem  37969  nninfnub  38346  eqab2  38845