Theorem riotaeqbidv

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011)

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

Proof

Step	Hyp	Ref	Expression
1		riotaeqbidv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		riotaeqbidv.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	2	riotabidv	⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜓 ) = ( ℩ 𝑥 ∈ 𝐴 𝜒 ) )
4	1	riotaeqdv	⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜒 ) = ( ℩ 𝑥 ∈ 𝐵 𝜒 ) )
5	3 4	eqtrd	⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜓 ) = ( ℩ 𝑥 ∈ 𝐵 𝜒 ) )