Theorem iuneq12df

Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016)

	Ref	Expression
Hypotheses	iuneq12df.1	⊢ Ⅎ 𝑥 𝜑
	iuneq12df.2	⊢ Ⅎ 𝑥 𝐴
	iuneq12df.3	⊢ Ⅎ 𝑥 𝐵
	iuneq12df.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	iuneq12df.5	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	iuneq12df	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		iuneq12df.1	⊢ Ⅎ 𝑥 𝜑
2		iuneq12df.2	⊢ Ⅎ 𝑥 𝐴
3		iuneq12df.3	⊢ Ⅎ 𝑥 𝐵
4		iuneq12df.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
5		iuneq12df.5	⊢ ( 𝜑 → 𝐶 = 𝐷 )
6	5	eleq2d	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷 ) )
7	1 2 3 4 6	rexeqbid	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) )
8	7	alrimiv	⊢ ( 𝜑 → ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) )
9		abbi1	⊢ ( ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 } )
10		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 }
11		df-iun	⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 }
12	9 10 11	3eqtr4g	⊢ ( ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )
13	8 12	syl	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )