Theorem iuneq12daf

Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017)

	Ref	Expression
Hypotheses	iuneq12daf.1	⊢ Ⅎ 𝑥 𝜑
	iuneq12daf.2	⊢ Ⅎ 𝑥 𝐴
	iuneq12daf.3	⊢ Ⅎ 𝑥 𝐵
	iuneq12daf.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	iuneq12daf.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 = 𝐷 )
Assertion	iuneq12daf	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		iuneq12daf.1	⊢ Ⅎ 𝑥 𝜑
2		iuneq12daf.2	⊢ Ⅎ 𝑥 𝐴
3		iuneq12daf.3	⊢ Ⅎ 𝑥 𝐵
4		iuneq12daf.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
5		iuneq12daf.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 = 𝐷 )
6	5	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷 ) )
7	1 6	rexbida	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ) )
8	2 3	rexeqf	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) )
9	4 8	syl	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) )
10	7 9	bitrd	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) )
11	10	alrimiv	⊢ ( 𝜑 → ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) )
12		abbi	⊢ ( ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 } )
13		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 }
14		df-iun	⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 }
15	12 13 14	3eqtr4g	⊢ ( ∀ 𝑦 ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐷 ) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )
16	11 15	syl	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )