Theorem iuneq12daf 32645

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

Hypotheses

Ref	Expression
iuneq12daf.1	⊢ Ⅎ𝑥𝜑
iuneq12daf.2	⊢ Ⅎ𝑥𝐴
iuneq12daf.3	⊢ Ⅎ𝑥𝐵
iuneq12daf.4	⊢ (𝜑 → 𝐴 = 𝐵)
iuneq12daf.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷)

Assertion

Ref	Expression
iuneq12daf	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Proof of Theorem iuneq12daf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq12daf.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
2		iuneq12daf.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷)
3	2	eleq2d 2825	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
4	1, 3	rexbida 3251	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐷))
5		iuneq12daf.4	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
6		iuneq12daf.2	. . . . . 6 ⊢ Ⅎ𝑥𝐴
7		iuneq12daf.3	. . . . . 6 ⊢ Ⅎ𝑥𝐵
8	6, 7	rexeqf 3321	. . . . 5 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
9	5, 8	syl 17	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
10	4, 9	bitrd 280	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
11	10	alrimiv 1934	. 2 ⊢ (𝜑 → ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
12		abbi 2804	. . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷})
13		df-iun 4923	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
14		df-iun 4923	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}
15	12, 13, 14	3eqtr4g 2799	. 2 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
16	11, 15	syl 17	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  {cab 2717  Ⅎwnfc 2886  ∃wrex 3063  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-iun 4923

This theorem is referenced by:  measvunilem0  34397