Theorem cbviuneq12df 40012

Description: Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)

Hypotheses

Ref	Expression
cbviuneq12df.xph	⊢ Ⅎ𝑥𝜑
cbviuneq12df.yph	⊢ Ⅎ𝑦𝜑
cbviuneq12df.x	⊢ Ⅎ𝑥𝑋
cbviuneq12df.y	⊢ Ⅎ𝑦𝑌
cbviuneq12df.xa	⊢ Ⅎ𝑥𝐴
cbviuneq12df.ya	⊢ Ⅎ𝑦𝐴
cbviuneq12df.b	⊢ Ⅎ𝑦𝐵
cbviuneq12df.xc	⊢ Ⅎ𝑥𝐶
cbviuneq12df.yc	⊢ Ⅎ𝑦𝐶
cbviuneq12df.d	⊢ Ⅎ𝑥𝐷
cbviuneq12df.f	⊢ Ⅎ𝑥𝐹
cbviuneq12df.g	⊢ Ⅎ𝑦𝐺
cbviuneq12df.xel	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴)
cbviuneq12df.yel	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)
cbviuneq12df.xsub	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹)
cbviuneq12df.ysub	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
cbviuneq12df.eq1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺)
cbviuneq12df.eq2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹)

Assertion

Ref	Expression
cbviuneq12df	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem cbviuneq12df

Step	Hyp	Ref	Expression
1		cbviuneq12df.xph	. . 3 ⊢ Ⅎ𝑥𝜑
2		cbviuneq12df.yph	. . 3 ⊢ Ⅎ𝑦𝜑
3		cbviuneq12df.y	. . 3 ⊢ Ⅎ𝑦𝑌
4		cbviuneq12df.ya	. . 3 ⊢ Ⅎ𝑦𝐴
5		cbviuneq12df.b	. . 3 ⊢ Ⅎ𝑦𝐵
6		cbviuneq12df.xc	. . 3 ⊢ Ⅎ𝑥𝐶
7		cbviuneq12df.yc	. . 3 ⊢ Ⅎ𝑦𝐶
8		cbviuneq12df.d	. . 3 ⊢ Ⅎ𝑥𝐷
9		cbviuneq12df.g	. . 3 ⊢ Ⅎ𝑦𝐺
10		cbviuneq12df.yel	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)
11		cbviuneq12df.ysub	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
12		cbviuneq12df.eq1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺)
13		eqimss 4026	. . . 4 ⊢ (𝐵 = 𝐺 → 𝐵 ⊆ 𝐺)
14	12, 13	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14	ss2iundf 40010	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
16		cbviuneq12df.x	. . 3 ⊢ Ⅎ𝑥𝑋
17		cbviuneq12df.xa	. . 3 ⊢ Ⅎ𝑥𝐴
18		cbviuneq12df.f	. . 3 ⊢ Ⅎ𝑥𝐹
19		cbviuneq12df.xel	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴)
20		cbviuneq12df.xsub	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹)
21		cbviuneq12df.eq2	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹)
22		eqimss 4026	. . . 4 ⊢ (𝐷 = 𝐹 → 𝐷 ⊆ 𝐹)
23	21, 22	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 ⊆ 𝐹)
24	2, 1, 16, 6, 8, 4, 17, 5, 18, 19, 20, 23	ss2iundf 40010	. 2 ⊢ (𝜑 → ∪ 𝑦 ∈ 𝐶 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
25	15, 24	eqssd 3987	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536  Ⅎwnf 1783   ∈ wcel 2113  Ⅎwnfc 2964   ⊆ wss 3939  ∪ ciun 4922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-v 3499  df-in 3946  df-ss 3955  df-iun 4924

This theorem is referenced by:  cbviuneq12dv  40013