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Theorem iuneq2 3985
 Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
iuneq2 (x A B = Cx A B = x A C)

Proof of Theorem iuneq2
StepHypRef Expression
1 ss2iun 3984 . . 3 (x A B Cx A B x A C)
2 ss2iun 3984 . . 3 (x A C Bx A C x A B)
31, 2anim12i 549 . 2 ((x A B C x A C B) → (x A B x A C x A C x A B))
4 eqss 3287 . . . 4 (B = C ↔ (B C C B))
54ralbii 2638 . . 3 (x A B = Cx A (B C C B))
6 r19.26 2746 . . 3 (x A (B C C B) ↔ (x A B C x A C B))
75, 6bitri 240 . 2 (x A B = C ↔ (x A B C x A C B))
8 eqss 3287 . 2 (x A B = x A C ↔ (x A B x A C x A C x A B))
93, 7, 83imtr4i 257 1 (x A B = Cx A B = x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ∀wral 2614   ⊆ wss 3257  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971 This theorem is referenced by:  iuneq2i  3987  iuneq2dv  3990
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