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Theorem cbviuneq12df 38270
 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 3690 . . . 4 (𝐵 = 𝐺𝐵𝐺)
1412, 13syl 17 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐺)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14ss2iundf 38268 . 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 3690 . . . 4 (𝐷 = 𝐹𝐷𝐹)
2321, 22syl 17 . . 3 ((𝜑𝑦𝐶) → 𝐷𝐹)
242, 1, 16, 6, 8, 4, 17, 5, 18, 19, 20, 23ss2iundf 38268 . 2 (𝜑 𝑦𝐶 𝐷 𝑥𝐴 𝐵)
2515, 24eqssd 3653 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 2780   ⊆ wss 3607  ∪ ciun 4552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-iun 4554 This theorem is referenced by:  cbviuneq12dv  38271
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