Theorem iuneq12d 4987

Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.) Remove DV conditions (Revised by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
iuneq12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
iuneq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
iuneq12d	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem iuneq12d

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq12d.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	eleq2d 2855	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	anbi1d 642	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶)))
4	3	rexbidv2 3191	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶))
5	4	abbidv 2835	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶})
6		df-iun 4959	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iun 4959	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2829	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
9		iuneq12d.2	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
10	9	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
11	10	iuneq2dv 4982	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
12	8, 11	eqtrd 2804	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095  ∪ ciun 4957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-v 3465  df-ss 3930  df-iun 4959

This theorem is referenced by:  disjiunb  5100  cfsmolem  10250  cfsmo  10251  wunex2  10719  wuncval2  10728  imasval  17561  lpival  21457  cnextval  24183  cnextfval  24184  dvfval  26021  fedgmullem1  33960  irngval  34016  mblfinlem2  38192  heiborlem10  38354  iunrelexpmin1  44321  iunrelexpmin2  44325  colleq12d  44850