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Theorem iuneq1 3670
 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 3668 . . 3 (𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
2 iunss1 3668 . . 3 (𝐵𝐴 𝑥𝐵 𝐶 𝑥𝐴 𝐶)
31, 2anim12i 321 . 2 ((𝐴𝐵𝐵𝐴) → ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶 𝑥𝐵 𝐶 𝑥𝐴 𝐶))
4 eqss 2960 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 2960 . 2 ( 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶 ↔ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶 𝑥𝐵 𝐶 𝑥𝐴 𝐶))
63, 4, 53imtr4i 190 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ⊆ wss 2917  ∪ ciun 3657 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659 This theorem is referenced by:  iuneq1d  3680  iununir  3738  iunsuc  4157  rdgisuc1  5971  rdg0  5974  oasuc  6044  omsuc  6051
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