Theorem cbviuneq12dv 43611

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
cbviuneq12dv.xel	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴)
cbviuneq12dv.yel	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)
cbviuneq12dv.xsub	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹)
cbviuneq12dv.ysub	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
cbviuneq12dv.eq1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺)
cbviuneq12dv.eq2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹)

Assertion

Ref	Expression
cbviuneq12dv	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)

Distinct variable groups:   𝑥,𝑦,𝜑   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷   𝑥,𝐹   𝑦,𝐺   𝑥,𝑋   𝑦,𝑌

Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑦)   𝐹(𝑦)   𝐺(𝑥)   𝑋(𝑦)   𝑌(𝑥)

Proof of Theorem cbviuneq12dv

Step	Hyp	Ref	Expression
1		nfv 1913	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1913	. 2 ⊢ Ⅎ𝑦𝜑
3		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝑋
4		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝑌
5		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝐴
6		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝐴
7		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝐵
8		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝐶
9		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝐶
10		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝐷
11		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝐹
12		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝐺
13		cbviuneq12dv.xel	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴)
14		cbviuneq12dv.yel	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)
15		cbviuneq12dv.xsub	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹)
16		cbviuneq12dv.ysub	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)
17		cbviuneq12dv.eq1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺)
18		cbviuneq12dv.eq2	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cbviuneq12df 43610	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∪ ciun 4964

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-v 3459  df-ss 3941  df-iun 4966

This theorem is referenced by:  trclfvdecomr  43677