Theorem ralrid 3060

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 3053	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylibr 234	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 2114  ∀wral 3052

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 207  df-ral 3053

This theorem is referenced by:  alral  3067  hbralrimi  3128  axprlem4  5373  ingru  10738  bnj1476  35022  bnj1533  35027  bnj1523  35246  r1omhfb  35287  r1omhfbregs  35312  bj-axnul  37311  bj-axreprepsep  37314  exrecfnlem  37623  nninfnub  37991  eqab2  38490