Theorem iuneq12daf 29373

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 2687	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
4	1, 3	rexbida 3047	. . . 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 3135	. . . . 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 268	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
11	10	alrimiv 1855	. 2 ⊢ (𝜑 → ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
12		abbi 2737	. . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷})
13		df-iun 4522	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
14		df-iun 4522	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}
15	13, 14	eqeq12i 2636	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷})
16	12, 15	bitr4i 267	. 2 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) ↔ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
17	11, 16	sylib 208	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  {cab 2608  Ⅎwnfc 2751  ∃wrex 2913  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-iun 4522

This theorem is referenced by:  measvunilem0  30276