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Theorem iuneq1 3717
Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iuneq1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iuneq1
StepHypRef Expression
1 iunss1 3715 . . 3 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
2 iunss1 3715 . . 3 (𝐵𝐴 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
31, 2anim12i 331 . 2 ((𝐴𝐵𝐵𝐴) → ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶 𝑥𝐵 𝐶 𝑥𝐴 𝐶))
4 eqss 3025 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3025 . 2 ( 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶 ↔ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶 𝑥𝐵 𝐶 𝑥𝐴 𝐶))
63, 4, 53imtr4i 199 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wss 2984   ciun 3704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-in 2990  df-ss 2997  df-iun 3706
This theorem is referenced by:  iuneq1d  3727  iununir  3785  iunsuc  4210  rdgisuc1  6079  rdg0  6082  oasuc  6155  omsuc  6163
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