Theorem riotaeqbidva

Description: Equivalent wff's yield equal restricted definition binders (deduction form). ( raleqbidva analog.) (Contributed by Thierry Arnoux, 29-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		riotaeqbidva.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		riotaeqbidva.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	riotabidva	⊢ ( 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 ) )