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Theorem iunxdif2 4504
 Description: Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
Hypothesis
Ref Expression
iunxdif2.1 (𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
iunxdif2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem iunxdif2
StepHypRef Expression
1 iunss2 4501 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷)
2 difss 3699 . . . . 5 (𝐴𝐵) ⊆ 𝐴
3 iunss1 4468 . . . . 5 ((𝐴𝐵) ⊆ 𝐴 𝑦 ∈ (𝐴𝐵)𝐷 𝑦𝐴 𝐷)
42, 3ax-mp 5 . . . 4 𝑦 ∈ (𝐴𝐵)𝐷 𝑦𝐴 𝐷
5 iunxdif2.1 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65cbviunv 4495 . . . 4 𝑥𝐴 𝐶 = 𝑦𝐴 𝐷
74, 6sseqtr4i 3601 . . 3 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶
81, 7jctil 558 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 → ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
9 eqss 3583 . 2 ( 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶 ↔ ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
108, 9sylibr 223 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  ∪ 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-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-dif 3543  df-in 3547  df-ss 3554  df-iun 4457 This theorem is referenced by: (None)
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