Theorem iuneq2f 36314

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 2920	. 2 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
5		eqidd 2739	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
6	3, 1, 2, 4, 5	iuneq12df 4950	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1539  Ⅎwnfc 2887  ∪ ciun 4924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rex 3070  df-iun 4926

This theorem is referenced by:  iuneq12f  36321