Theorem iuneq12f 38662

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 4969	. 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 38655	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷)
5	1, 4	sylan9eqr 2819	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  Ⅎwnfc 2909  ∀wral 3076  ∪ ciun 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-v 3456  df-ss 3921  df-iun 4951

This theorem is referenced by: (None)