Theorem iuneq2f 38695

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 2944	. 2 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		id 23	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
5		eqidd 2770	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
6	3, 1, 2, 4, 5	iuneq12df 4987	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1567  Ⅎwnfc 2916  ∪ ciun 4960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-iun 4962

This theorem is referenced by:  iuneq12f  38702