Theorem cbviuneq12dv 41159

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 1918	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1918	. 2 ⊢ Ⅎ𝑦𝜑
3		nfcv 2906	. 2 ⊢ Ⅎ𝑥𝑋
4		nfcv 2906	. 2 ⊢ Ⅎ𝑦𝑌
5		nfcv 2906	. 2 ⊢ Ⅎ𝑥𝐴
6		nfcv 2906	. 2 ⊢ Ⅎ𝑦𝐴
7		nfcv 2906	. 2 ⊢ Ⅎ𝑦𝐵
8		nfcv 2906	. 2 ⊢ Ⅎ𝑥𝐶
9		nfcv 2906	. 2 ⊢ Ⅎ𝑦𝐶
10		nfcv 2906	. 2 ⊢ Ⅎ𝑥𝐷
11		nfcv 2906	. 2 ⊢ Ⅎ𝑥𝐹
12		nfcv 2906	. 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 41158	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-v 3424  df-in 3890  df-ss 3900  df-iun 4923

This theorem is referenced by:  trclfvdecomr  41225