Theorem iuneq12d 4979

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 2846	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	anbi1d 643	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶)))
4	3	rexbidv2 3182	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶))
5	4	abbidv 2826	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶})
6		df-iun 4952	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iun 4952	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2820	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
9		iuneq12d.2	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
10	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
11	10	iuneq2dv 4975	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
12	8, 11	eqtrd 2795	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  ∪ ciun 4950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3897  df-iun 4952

This theorem is used by:  disjiunb  5092  cfsmolem  10355  cfsmo  10356  wunex2  10830  wuncval2  10839  imasval  17692  lpival  21657  cnextval  24389  cnextfval  24390  dvfval  26226  fedgmullem1  34311  irngval  34367  mblfinlem2  38685  heiborlem10  38863  iunrelexpmin1  44817  iunrelexpmin2  44821  colleq12d  45346