Theorem iuneq12f 36058

Description: Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Hypotheses

Ref	Expression
iuneq12f.1	⊢ Ⅎ𝑥𝐴
iuneq12f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
iuneq12f	⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Proof of Theorem iuneq12f

Step	Hyp	Ref	Expression
1		iuneq2 4923	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐴 𝐷)
2		iuneq12f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		iuneq12f.2	. . 3 ⊢ Ⅎ𝑥𝐵
4	2, 3	iuneq2f 36051	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷)
5	1, 4	sylan9eqr 2800	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543  Ⅎwnfc 2884  ∀wral 3061  ∪ ciun 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-v 3410  df-in 3873  df-ss 3883  df-iun 4906

This theorem is referenced by: (None)