Theorem raleqbid

Description: Equality deduction for restricted universal quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017)

	Ref	Expression
Hypotheses	raleqbid.0	⊢ Ⅎ 𝑥 𝜑
	raleqbid.1	⊢ Ⅎ 𝑥 𝐴
	raleqbid.2	⊢ Ⅎ 𝑥 𝐵
	raleqbid.3	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	raleqbid.4	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	raleqbid	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		raleqbid.0	⊢ Ⅎ 𝑥 𝜑
2		raleqbid.1	⊢ Ⅎ 𝑥 𝐴
3		raleqbid.2	⊢ Ⅎ 𝑥 𝐵
4		raleqbid.3	⊢ ( 𝜑 → 𝐴 = 𝐵 )
5		raleqbid.4	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
6	2 3	raleqf	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )
7	4 6	syl	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )
8	1 5	ralbid	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) )
9	7 8	bitrd	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜒 ) )