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Theorem cbviuneq12df 41269
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 𝑥𝜑
cbviuneq12df.yph 𝑦𝜑
cbviuneq12df.x 𝑥𝑋
cbviuneq12df.y 𝑦𝑌
cbviuneq12df.xa 𝑥𝐴
cbviuneq12df.ya 𝑦𝐴
cbviuneq12df.b 𝑦𝐵
cbviuneq12df.xc 𝑥𝐶
cbviuneq12df.yc 𝑦𝐶
cbviuneq12df.d 𝑥𝐷
cbviuneq12df.f 𝑥𝐹
cbviuneq12df.g 𝑦𝐺
cbviuneq12df.xel ((𝜑𝑦𝐶) → 𝑋𝐴)
cbviuneq12df.yel ((𝜑𝑥𝐴) → 𝑌𝐶)
cbviuneq12df.xsub ((𝜑𝑦𝐶𝑥 = 𝑋) → 𝐵 = 𝐹)
cbviuneq12df.ysub ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)
cbviuneq12df.eq1 ((𝜑𝑥𝐴) → 𝐵 = 𝐺)
cbviuneq12df.eq2 ((𝜑𝑦𝐶) → 𝐷 = 𝐹)
Assertion
Ref Expression
cbviuneq12df (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 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
StepHypRef Expression
1 cbviuneq12df.xph . . 3 𝑥𝜑
2 cbviuneq12df.yph . . 3 𝑦𝜑
3 cbviuneq12df.y . . 3 𝑦𝑌
4 cbviuneq12df.ya . . 3 𝑦𝐴
5 cbviuneq12df.b . . 3 𝑦𝐵
6 cbviuneq12df.xc . . 3 𝑥𝐶
7 cbviuneq12df.yc . . 3 𝑦𝐶
8 cbviuneq12df.d . . 3 𝑥𝐷
9 cbviuneq12df.g . . 3 𝑦𝐺
10 cbviuneq12df.yel . . 3 ((𝜑𝑥𝐴) → 𝑌𝐶)
11 cbviuneq12df.ysub . . 3 ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)
12 cbviuneq12df.eq1 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐺)
13 eqimss 3977 . . . 4 (𝐵 = 𝐺𝐵𝐺)
1412, 13syl 17 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐺)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14ss2iundf 41267 . 2 (𝜑 𝑥𝐴 𝐵 𝑦𝐶 𝐷)
16 cbviuneq12df.x . . 3 𝑥𝑋
17 cbviuneq12df.xa . . 3 𝑥𝐴
18 cbviuneq12df.f . . 3 𝑥𝐹
19 cbviuneq12df.xel . . 3 ((𝜑𝑦𝐶) → 𝑋𝐴)
20 cbviuneq12df.xsub . . 3 ((𝜑𝑦𝐶𝑥 = 𝑋) → 𝐵 = 𝐹)
21 cbviuneq12df.eq2 . . . 4 ((𝜑𝑦𝐶) → 𝐷 = 𝐹)
22 eqimss 3977 . . . 4 (𝐷 = 𝐹𝐷𝐹)
2321, 22syl 17 . . 3 ((𝜑𝑦𝐶) → 𝐷𝐹)
242, 1, 16, 6, 8, 4, 17, 5, 18, 19, 20, 23ss2iundf 41267 . 2 (𝜑 𝑦𝐶 𝐷 𝑥𝐴 𝐵)
2515, 24eqssd 3938 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wnf 1786  wcel 2106  wnfc 2887  wss 3887   ciun 4924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-in 3894  df-ss 3904  df-iun 4926
This theorem is referenced by:  cbviuneq12dv  41270
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