Theorem iuneq12f 38164

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 5019	. 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 38157	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷)
5	1, 4	sylan9eqr 2799	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  Ⅎwnfc 2890  ∀wral 3061  ∪ ciun 4999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3483  df-ss 3983  df-iun 5001

This theorem is referenced by: (None)