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Theorem iunxdif2 3696
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 3693 . . 3 (x A y (AB)𝐶𝐷 x A 𝐶 y (AB)𝐷)
2 difss 3064 . . . . 5 (AB) ⊆ A
3 iunss1 3659 . . . . 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 3687 . . . 4 x A 𝐶 = y A 𝐷
74, 6sseqtr4i 2972 . . 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 2954 . 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 1242  wral 2300  wrex 2301  cdif 2908  wss 2911   ciun 3648
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-iun 3650
This theorem is referenced by: (None)
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