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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbviuneq12dv | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
cbviuneq12dv | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1843 | . 2 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝑋 | |
4 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝑌 | |
5 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐴 | |
6 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐴 | |
7 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐵 | |
8 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐶 | |
9 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐶 | |
10 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐷 | |
11 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2764 | . 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 37953 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∪ ciun 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-in 3581 df-ss 3588 df-iun 4522 |
This theorem is referenced by: trclfvdecomr 38020 |
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