Theorem ralbidv2

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997)

	Ref	Expression
Hypothesis	ralbidv2.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 → 𝜒 ) ) )
Assertion	ralbidv2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralbidv2.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 → 𝜒 ) ) )
2	1	albidv	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜒 ) ) )
3		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
4		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 𝜒 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜒 ) )
5	2 3 4	3bitr4g	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) )