Theorem rabrabiOLD

Description: Obsolete version of rabrabi as of 12-Oct-2024. (Contributed by AV, 2-Aug-2022) Avoid ax-10 and ax-11 . (Revised by Gino Giotto, 20-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		rabrabiOLD.1	$$⊢x=y\to \left(\chi \leftrightarrow \phi \right)$$
2	1	cbvrabv	$$⊢\left\{x\in A|\chi \right\}=\left\{y\in A|\phi \right\}$$
3	2	rabeqi	$$⊢\left\{x\in \left\{x\in A|\chi \right\}|\psi \right\}=\left\{x\in \left\{y\in A|\phi \right\}|\psi \right\}$$
4		rabrab	$$⊢\left\{x\in \left\{x\in A|\chi \right\}|\psi \right\}=\left\{x\in A|\left(\chi \wedge \psi \right)\right\}$$
5	3 4	eqtr3i	$$⊢\left\{x\in \left\{y\in A|\phi \right\}|\psi \right\}=\left\{x\in A|\left(\chi \wedge \psi \right)\right\}$$