Theorem expandral 41371

Description: Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypothesis

Ref	Expression
expandral.1	⊢ (𝜑 ↔ 𝜓)

Assertion

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

Proof of Theorem expandral

Step	Hyp	Ref	Expression
1		expandral.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	ralbii 3097	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)
3		df-ral 3075	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4	2, 3	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  ∀wral 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 210  df-ral 3075

This theorem is referenced by:  expanduniss  41374  ismnuprim  41375