Theorem r19.21v

Description: Restricted quantifier version of 19.21v . (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020)

	Ref	Expression
Assertion	r19.21v	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bi2.04	⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) ↔ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
2	1	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) ↔ ∀ 𝑥 ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
3		19.21v	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
5		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
6		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
7	6	imbi2i	⊢ ( ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
8	4 5 7	3bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) )