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Theorem cbviuneq12dv 36973
 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 ((𝜑𝑦𝐶) → 𝑋𝐴)
cbviuneq12dv.yel ((𝜑𝑥𝐴) → 𝑌𝐶)
cbviuneq12dv.xsub ((𝜑𝑦𝐶𝑥 = 𝑋) → 𝐵 = 𝐹)
cbviuneq12dv.ysub ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)
cbviuneq12dv.eq1 ((𝜑𝑥𝐴) → 𝐵 = 𝐺)
cbviuneq12dv.eq2 ((𝜑𝑦𝐶) → 𝐷 = 𝐹)
Assertion
Ref Expression
cbviuneq12dv (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 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
StepHypRef Expression
1 nfv 1830 . 2 𝑥𝜑
2 nfv 1830 . 2 𝑦𝜑
3 nfcv 2751 . 2 𝑥𝑋
4 nfcv 2751 . 2 𝑦𝑌
5 nfcv 2751 . 2 𝑥𝐴
6 nfcv 2751 . 2 𝑦𝐴
7 nfcv 2751 . 2 𝑦𝐵
8 nfcv 2751 . 2 𝑥𝐶
9 nfcv 2751 . 2 𝑦𝐶
10 nfcv 2751 . 2 𝑥𝐷
11 nfcv 2751 . 2 𝑥𝐹
12 nfcv 2751 . 2 𝑦𝐺
13 cbviuneq12dv.xel . 2 ((𝜑𝑦𝐶) → 𝑋𝐴)
14 cbviuneq12dv.yel . 2 ((𝜑𝑥𝐴) → 𝑌𝐶)
15 cbviuneq12dv.xsub . 2 ((𝜑𝑦𝐶𝑥 = 𝑋) → 𝐵 = 𝐹)
16 cbviuneq12dv.ysub . 2 ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)
17 cbviuneq12dv.eq1 . 2 ((𝜑𝑥𝐴) → 𝐵 = 𝐺)
18 cbviuneq12dv.eq2 . 2 ((𝜑𝑦𝐶) → 𝐷 = 𝐹)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18cbviuneq12df 36972 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-iun 4457 This theorem is referenced by:  trclfvdecomr  37039
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