Theorem elrabrd

Description: Deduction version of elrab , just like elrabd , but backwards direction. (Contributed by Thierry Arnoux, 15-Jan-2026)

	Ref	Expression
Hypotheses	elrabrd.1	$$⊢x=A\to \left(\psi \leftrightarrow \chi \right)$$
	elrabrd.2	$$⊢\phi \to A\in \left\{x\in B|\psi \right\}$$
Assertion	elrabrd	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		elrabrd.1	$$⊢x=A\to \left(\psi \leftrightarrow \chi \right)$$
2		elrabrd.2	$$⊢\phi \to A\in \left\{x\in B|\psi \right\}$$
3	1	elrab	$$⊢A\in \left\{x\in B|\psi \right\}\leftrightarrow \left(A\in B\wedge \chi \right)$$
4	2 3	sylib	$$⊢\phi \to \left(A\in B\wedge \chi \right)$$
5	4	simprd	$$⊢\phi \to \chi$$