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Theorem iuneq2 3647
Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
iuneq2 (x A B = 𝐶 x A B = x A 𝐶)

Proof of Theorem iuneq2
StepHypRef Expression
1 ss2iun 3646 . . 3 (x A B𝐶 x A B x A 𝐶)
2 ss2iun 3646 . . 3 (x A 𝐶B x A 𝐶 x A B)
31, 2anim12i 321 . 2 ((x A B𝐶 x A 𝐶B) → ( x A B x A 𝐶 x A 𝐶 x A B))
4 eqss 2937 . . . 4 (B = 𝐶 ↔ (B𝐶 𝐶B))
54ralbii 2308 . . 3 (x A B = 𝐶x A (B𝐶 𝐶B))
6 r19.26 2419 . . 3 (x A (B𝐶 𝐶B) ↔ (x A B𝐶 x A 𝐶B))
75, 6bitri 173 . 2 (x A B = 𝐶 ↔ (x A B𝐶 x A 𝐶B))
8 eqss 2937 . 2 ( x A B = x A 𝐶 ↔ ( x A B x A 𝐶 x A 𝐶 x A B))
93, 7, 83imtr4i 190 1 (x A B = 𝐶 x A B = x A 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228  wral 2284  wss 2894   ciun 3631
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-in 2901  df-ss 2908  df-iun 3633
This theorem is referenced by:  iuneq2i  3649  iuneq2dv  3652  dfmptg  5267
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