Theorem ralbida

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003)

	Ref	Expression
Hypotheses	ralbida.1	⊢ Ⅎ 𝑥 𝜑
	ralbida.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	ralbida	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralbida.1	⊢ Ⅎ 𝑥 𝜑
2		ralbida.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	pm5.74da	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
4	1 3	albid	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
5		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
6		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜒 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜒 ) )
7	4 5 6	3bitr4g	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 𝜒 ) )