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Theorem iunxdif2 3675
 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 (x A y (AB)𝐶𝐷 y (AB)𝐷 = x A 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 3672 . . 3 (x A y (AB)𝐶𝐷 x A 𝐶 y (AB)𝐷)
2 difss 3043 . . . . 5 (AB) ⊆ A
3 iunss1 3638 . . . . 5 ((AB) ⊆ A y (AB)𝐷 y A 𝐷)
42, 3ax-mp 7 . . . 4 y (AB)𝐷 y A 𝐷
5 iunxdif2.1 . . . . 5 (x = y𝐶 = 𝐷)
65cbviunv 3666 . . . 4 x A 𝐶 = y A 𝐷
74, 6sseqtr4i 2951 . . 3 y (AB)𝐷 x A 𝐶
81, 7jctil 295 . 2 (x A y (AB)𝐶𝐷 → ( y (AB)𝐷 x A 𝐶 x A 𝐶 y (AB)𝐷))
9 eqss 2933 . 2 ( y (AB)𝐷 = x A 𝐶 ↔ ( y (AB)𝐷 x A 𝐶 x A 𝐶 y (AB)𝐷))
108, 9sylibr 137 1 (x A y (AB)𝐶𝐷 y (AB)𝐷 = x A 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226  ∀wral 2280  ∃wrex 2281   ∖ cdif 2887   ⊆ wss 2890  ∪ ciun 3627 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-in 2897  df-ss 2904  df-iun 3629 This theorem is referenced by: (None)
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