Theorem 3unrab

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

	Ref	Expression
Assertion	3unrab	⊢ ( ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐴 ∣ 𝜓 } ) ∪ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ) = { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) }

Proof

Step	Hyp	Ref	Expression
1		unrab	⊢ ( { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∨ 𝜓 ) } ∪ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ) = { 𝑥 ∈ 𝐴 ∣ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) }
2		unrab	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐴 ∣ 𝜓 } ) = { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∨ 𝜓 ) }
3	2	uneq1i	⊢ ( ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐴 ∣ 𝜓 } ) ∪ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ) = ( { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∨ 𝜓 ) } ∪ { 𝑥 ∈ 𝐴 ∣ 𝜒 } )
4		df-3or	⊢ ( ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) )
5	4	rabbii	⊢ { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) } = { 𝑥 ∈ 𝐴 ∣ ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) }
6	1 3 5	3eqtr4i	⊢ ( ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐴 ∣ 𝜓 } ) ∪ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ) = { 𝑥 ∈ 𝐴 ∣ ( 𝜑 ∨ 𝜓 ∨ 𝜒 ) }