Theorem unieqOLD

Description: Obsolete version of unieq as of 13-Apr-2024. (Contributed by NM, 10-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.) 29-Jun-2011.)

	Ref	Expression
Assertion	unieqOLD	⊢ ( 𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rexeq	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ) )
2	1	abbidv	⊢ ( 𝐴 = 𝐵 → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 } )
3		dfuni2	⊢ ∪ 𝐴 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 }
4		dfuni2	⊢ ∪ 𝐵 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 }
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵 )