Theorem iuneq12dOLD 5024

Description: Obsolete version of iuneq12d 5025 as of 1-Sep-2025. (Contributed by Drahflow, 22-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
iuneq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
iuneq12dOLD.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
iuneq12dOLD	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

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

Proof of Theorem iuneq12dOLD

Step	Hyp	Ref	Expression
1		iuneq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	iuneq1d 5023	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
3		iuneq12dOLD.2	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
5	4	iuneq2dv 5020	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
6	2, 5	eqtrd 2774	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105  ∪ ciun 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-ss 3979  df-iun 4997

This theorem is referenced by: (None)