Theorem iuneq2f 32999

Description: Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Hypotheses

Ref	Expression
iuneq2f.1	⊢ Ⅎ𝑥𝐴
iuneq2f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
iuneq2f	⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Proof of Theorem iuneq2f

Step	Hyp	Ref	Expression
1		iuneq2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		iuneq2f.2	. . 3 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2666	. 2 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
5		eqidd 2515	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
6	3, 1, 2, 4, 5	iuneq12df 4378	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1474  Ⅎwnfc 2642  ∪ ciun 4353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494

This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rex 2806  df-iun 4355

This theorem is referenced by:  iuneq12f  33008