Theorem abbi

Description: Equivalent formulas yield equal class abstractions (closed form). This is the backward implication of abbib , proved from fewer axioms, and hence is independently named. (Contributed by BJ and WL and SN, 20-Aug-2023)

	Ref	Expression
Assertion	abbi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left\{x|\phi \right\}=\left\{x|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		spsbbi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\left[y/x\right]\phi \leftrightarrow \left[y/x\right]\psi \right)$$
2		df-clab	$$⊢y\in \left\{x|\phi \right\}\leftrightarrow \left[y/x\right]\phi$$
3		df-clab	$$⊢y\in \left\{x|\psi \right\}\leftrightarrow \left[y/x\right]\psi$$
4	1 2 3	3bitr4g	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(y\in \left\{x|\phi \right\}\leftrightarrow y\in \left\{x|\psi \right\}\right)$$
5	4	eqrdv	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left\{x|\phi \right\}=\left\{x|\psi \right\}$$