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Theorem iuneq1 3982
 Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iuneq1 (A = Bx A C = x B C)
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   C(x)

Proof of Theorem iuneq1
StepHypRef Expression
1 iunss1 3980 . . 3 (A Bx A C x B C)
2 iunss1 3980 . . 3 (B Ax B C x A C)
31, 2anim12i 549 . 2 ((A B B A) → (x A C x B C x B C x A C))
4 eqss 3287 . 2 (A = B ↔ (A B B A))
5 eqss 3287 . 2 (x A C = x B C ↔ (x A C x B C x B C x A C))
63, 4, 53imtr4i 257 1 (A = Bx A C = x B C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3257  ∪ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  iuneq1d  3992  iununi  4050
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