Theorem ralrid 3084

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 3077	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylibr 237	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   ∈ wcel 2144  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ral 3077

This theorem is used by:  alral  3091  hbralrimi  3152  axprlem4  5387  ingru  10900  bnj1476  35519  bnj1533  35524  bnj1523  35743  r1omhfb  35780  r1omhfbregs  35847  bj-axnul  38030  bj-axreprepsep  38033  exrecfnlem  38345  nninfnub  38728  eqab2  39225