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Theorem iunxdif2 4968
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 4964 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷)
2 difss 4105 . . . . 5 (𝐴𝐵) ⊆ 𝐴
3 iunss1 4924 . . . . 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 4956 . . . 4 𝑥𝐴 𝐶 = 𝑦𝐴 𝐷
74, 6sseqtrri 4001 . . 3 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶
81, 7jctil 520 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 → ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
9 eqss 3979 . 2 ( 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶 ↔ ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
108, 9sylibr 235 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wral 3135  wrex 3136  cdif 3930  wss 3933   ciun 4910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-iun 4912
This theorem is referenced by: (None)
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