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	$$⊢\phi \to A=B$$
	riotaeqbidva.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	riotaeqbidva	$$⊢\phi \to \left(\iota x\in A|\psi \right)=\left(\iota x\in B|\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		riotaeqbidva.1	$$⊢\phi \to A=B$$
2		riotaeqbidva.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	riotabidva	$$⊢\phi \to \left(\iota x\in A|\psi \right)=\left(\iota x\in A|\chi \right)$$
4	1	riotaeqdv	$$⊢\phi \to \left(\iota x\in A|\chi \right)=\left(\iota x\in B|\chi \right)$$
5	3 4	eqtrd	$$⊢\phi \to \left(\iota x\in A|\psi \right)=\left(\iota x\in B|\chi \right)$$