Theorem iuneq12d 4951

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 2825	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	anbi1d 637	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶)))
4	3	rexbidv2 3159	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶))
5	4	abbidv 2805	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶})
6		df-iun 4923	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iun 4923	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2799	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
9		iuneq12d.2	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
10	9	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
11	10	iuneq2dv 4946	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
12	8, 11	eqtrd 2774	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {cab 2717  ∃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-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-v 3433  df-ss 3900  df-iun 4923

This theorem is referenced by:  disjiunb  5062  cfsmolem  10183  cfsmo  10184  wunex2  10652  wuncval2  10661  imasval  17466  lpival  21317  cnextval  24044  cnextfval  24045  dvfval  25882  fedgmullem1  33813  irngval  33869  mblfinlem2  38025  heiborlem10  38187  iunrelexpmin1  44152  iunrelexpmin2  44156  colleq12d  44697