Theorem rabbidvaOLD

Description: Obsolete proof of rabbidva as of 4-Dec-2023. (Contributed by NM, 28-Nov-2003) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	rabbidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rabbidvaOLD	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in A|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabbidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	ralrimiva	$$⊢\phi \to \forall x\in A\left(\psi \leftrightarrow \chi \right)$$
3		rabbi	$$⊢\forall x\in A\left(\psi \leftrightarrow \chi \right)\leftrightarrow \left\{x\in A|\psi \right\}=\left\{x\in A|\chi \right\}$$
4	2 3	sylib	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in A|\chi \right\}$$