Theorem raleqbidva

Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		raleqbidva.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		raleqbidva.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	ralbidva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 𝜒 ) )
4	1	raleqdv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜒 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) )