Theorem rabrabi

Description: Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022) Avoid ax-10 and ax-11 . (Revised by Gino Giotto, 20-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		rabrabi.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	$$⊢\left\{x\in \left\{y\in A|\phi \right\}|\psi \right\}=\left\{x\in A|\left(\chi \wedge \psi \right)\right\}$$