Theorem rabidd

Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of Quine p. 16. (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	rabidd.1	$$⊢\phi \to x\in A$$
	rabidd.2	$$⊢\phi \to \chi$$
Assertion	rabidd	$$⊢\phi \to x\in \left\{x\in A|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabidd.1	$$⊢\phi \to x\in A$$
2		rabidd.2	$$⊢\phi \to \chi$$
3		rabid	$$⊢x\in \left\{x\in A|\chi \right\}\leftrightarrow \left(x\in A\wedge \chi \right)$$
4	1 2 3	sylanbrc	$$⊢\phi \to x\in \left\{x\in A|\chi \right\}$$