Theorem 3unrab

Description: Union of three restricted class abstractions. (Contributed by Thierry Arnoux, 6-Jul-2025)

	Ref	Expression
Assertion	3unrab	$$⊢\left(\left\{x\in A|\phi \right\}\cup \left\{x\in A|\psi \right\}\right)\cup \left\{x\in A|\chi \right\}=\left\{x\in A|\left(\phi \vee \psi \vee \chi \right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		unrab	$$⊢\left\{x\in A|\left(\phi \vee \psi \right)\right\}\cup \left\{x\in A|\chi \right\}=\left\{x\in A|\left(\left(\phi \vee \psi \right)\vee \chi \right)\right\}$$
2		unrab	$$⊢\left\{x\in A|\phi \right\}\cup \left\{x\in A|\psi \right\}=\left\{x\in A|\left(\phi \vee \psi \right)\right\}$$
3	2	uneq1i	$$⊢\left(\left\{x\in A|\phi \right\}\cup \left\{x\in A|\psi \right\}\right)\cup \left\{x\in A|\chi \right\}=\left\{x\in A|\left(\phi \vee \psi \right)\right\}\cup \left\{x\in A|\chi \right\}$$
4		df-3or	$$⊢\left(\phi \vee \psi \vee \chi \right)\leftrightarrow \left(\left(\phi \vee \psi \right)\vee \chi \right)$$
5	4	rabbii	$$⊢\left\{x\in A|\left(\phi \vee \psi \vee \chi \right)\right\}=\left\{x\in A|\left(\left(\phi \vee \psi \right)\vee \chi \right)\right\}$$
6	1 3 5	3eqtr4i	$$⊢\left(\left\{x\in A|\phi \right\}\cup \left\{x\in A|\psi \right\}\right)\cup \left\{x\in A|\chi \right\}=\left\{x\in A|\left(\phi \vee \psi \vee \chi \right)\right\}$$